Coend coalgebras and reconstruction from finite fibres

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

Finite-dimensional observations can recover a category without a projective generator or a finite list of simple objects. The information they recover is naturally a coalgebra. We construct its matrix coefficients, prove that they recover every object and morphism, and identify the completed algebra of natural operators on the observations.

Let \(k\) be any field. Let \(A\) be an essentially small \(k\)-linear abelian category and

an exact faithful \(k\)-linear functor. Choose a small equivalent category when forming sums over its objects and arrows. All vector spaces, diagrams and tensor products belong to fixed universes, and tensors are over \(k\). No monoidal structure on \(A\) or on \(F\) is assumed.

The result is a coalgebra \(C_F\), together with equivalences over the indicated underlying vector-space functors:

Comodules are right comodules. Their coaction is \(M\to M\otimes C_F\).

Use Ind-objects through their elements, Section 3, and Two cokernels and abelian completion, Sections 2–6, for the exact constant embedding, common arrow models, exact Ind extension and all small colimits. Presentable probes, Section 7 proves the cocontinuous adjoint criterion used below. Finite subcoalgebras, Section 1 proves the finite-subcomodule theorem. Comodules and rational modules, Sections 1–5 gives the abelian comodule category, finite-piece realization and actual cofree adjunction. These are proved prerequisites; reconstruction from \(F\) will be proved here. A classical account of the reconstruction statement is [EGNO], Section 1.10.

1. The coefficient quotient and its coalgebra

Form the vector space

Here \(R\) is the span of the differences in the second line; brackets name the corresponding quotient classes. This is a small sum with a small set of relations. Its universal property is that linear maps out of \(C_F\) are families of bilinear coefficient maps respecting exactly those relations.

For a basis \(e_i\) of \(F(a)\), with dual basis \(e_i^*\), prescribe

The tensor \(\sum_i e_i\otimes e_i^*\) corresponds to the identity endomorphism of \(F(a)\), so is basis independent. Thus the proposed coproduct is basis independent before imposing any relations.

Lemma 1.1. These formulas descend and make \(C_F\) a coalgebra.

Proof: the relations. Let \(e_i\) be a basis of \(F(a)\) and \(w_j\) a basis of \(F(b)\). In the coproduct of \([\psi\otimes F(f)v]_b\), use (1.1) on the second tensor factor to obtain

In the coproduct of \([\psi F(f)\otimes v]_a\), use (1.1) on the first factor, expand \(F(f)e_i\) in the \(w_j\), and collect terms. The second factor becomes the functional \(\sum_i w_j^*(F(f)e_i)e_i^*=w_j^*F(f)\), so the result is (1.3). Thus the coproduct respects every generating relation. The two counit values both equal \(\psi(F(f)v)\), so it descends as well.

Proof: the identities. Applying either iterated coproduct to a coefficient gives, after renaming the two finite indices,

This proves coassociativity. Applying the counit in the first factor of (1.2) uses \(\sum_i\phi(e_i)e_i^*=\phi\); applying it in the second uses \(\sum_i e_i e_i^*(v)=v\). Both give the original coefficient. Since coefficients span the quotient, both counit identities hold. \(\square\)

There is a canonical coaction on every fibre:

It is basis independent by the same identity tensor. Equations (1.2) give its coassociativity and counit. For \(f:a\to b\), expanding in the two bases and using (1.1) gives \((F(f)\otimes1)\rho_a=\rho_bF(f)\). Thus \(F\) lifts to a \(k\)-linear functor \(K_0:A\to\operatorname{Comod}^{\rm fd}(C_F)\).

The remaining task is to prove that this particular functor is an equivalence. Defining the coefficient coalgebra alone does not prove that assertion.

2. Extend the observations to the abelian Ind category

Put \(B=\operatorname{Ind}(A)\), with constant objects still denoted by \(a\). Extend \(F\) by

The Ind Hom formula defines it on every arrow by applying \(F\) to stage representatives and then using the target colimit. Eventual equality makes this independent of representatives, and the same formula preserves identities, composition and \(k\)-linear combinations. This is the extension proved in the Ind prerequisite. Its abelian common-arrow theorem and exactness of \(F\) make \(U\) exact: kernels and cokernels are computed at a common filtered stage, and the vector-space filtered colimit preserves those exact comparisons.

It preserves small coproducts. A coproduct is the filtered colimit of finite sums of its entries, and \(U\) preserves those finite sums and filtered colimits. Together with cokernel preservation, the usual sum-and-relations construction of a diagram colimit shows that \(U\) preserves all small colimits. These are actual comparisons with the specified colimit maps.

Lemma 2.1. The functor \(U\) is faithful and reflects isomorphisms.

Proof. First suppose \(U(X)=0\), with \(X\) presented as in (2.1). For a map \(h:a\to X\), compactness of the constant \(a\) represents it by \(h_j:a\to a_j\) at a stage. The map \(F(a)\to\operatorname{colim}_jF(a_j)\) is zero. A basis of \(F(a)\) is finite, so its finitely many zero witnesses occur at a common later stage. The resulting stage composite of \(F(h_j)\) is zero. Faithfulness of \(F\) makes the corresponding composite in \(A\) zero; thus \(h=0\).

The constants jointly detect objects: \(X\), as its represented filtered-colimit presheaf, is zero when all the groups \(\operatorname{Hom}_B(a,X)\) are zero. Hence \(X=0\). If \(U(g)=0\), exactness gives \(U(\operatorname{im}g)=0\), so \(\operatorname{im}g=0\) and \(g=0\). Additivity now proves faithfulness on each Hom space. Finally, if \(U(g)\) is invertible, exactness makes its kernel and cokernel zero under \(U\), hence zero in \(B\). The abelian coimage-to-image theorem makes \(g\) invertible. \(\square\)

The category \(B\) is locally finitely presentable: it has all small colimits by the abelian completion theorem; its small constant family consists of finitely presentable objects by the Ind Hom formula; and every object is a filtered colimit of that family by definition. The cocontinuous adjoint criterion therefore gives a right adjoint

Write \(\eta:1_B\to RU\) for the unit and \(\varepsilon:UR\to1\) for the counit. The adjunction's actual Hom bijection is \(k\)-linear: its forward map sends \(h:X\to R(V)\) to \(\varepsilon_VU(h)\), which is linear in \(h\). Consequently \(R\) is \(k\)-linear as well, since its action on maps is tested through these linear bijections.

3. The right adjoint is cocontinuous and gives a coalgebra

Lemma 3.1. The functor \(R\) preserves all small colimits.

Proof. Every short exact vector-space sequence splits, by extending a basis. The additive functor \(R\) preserves its splitting equations and finite biproducts, so the image row is split exact in \(B\). Thus \(R\) is exact.

For a small family \(V_s\), test the canonical map \(\bigoplus_sR(V_s)\to R(\bigoplus_sV_s)\) against a constant \(a\). Compactness of \(a\), the description of a sum by finite partial sums, and the adjunction identify its Hom map with

It is bijective: the images of finitely many basis vectors have supports contained in one finite set, and equality is componentwise. The constants jointly reflect isomorphisms, because their Hom values are the Ind presheaf itself. Hence the canonical sum map is invertible. Exactness supplies cokernels, and coproducts with cokernels supply all small colimits. \(\square\)

Let \(Q=UR\), with comultiplication \(\delta=U\eta R\) and counit \(\varepsilon\); the adjunction identities give its comonad identities. These are also direct consequences of naturality of \(\eta\) and the two triangles. Put \(H=Q(k)\). There is a canonical natural isomorphism

Linearity of \(Q\) makes the rule bilinear. A basis decomposition of \(V\) into a coproduct of copies of \(k\), together with coproduct preservation, proves that it is invertible. Its definition proves naturality without choosing bases naturally.

Transport \(\delta_k\) by \(\alpha_H^{-1}\) to \(\Delta_H:H\to H\otimes H\), and put \(\epsilon_H=\varepsilon_k:H\to k\). Naturality at all \(\ell_v\) says that under (3.2), \(\delta_V\) is \(1_V\otimes\Delta_H\) and \(\varepsilon_V\) is \(1_V\otimes\epsilon_H\). Thus the comonad identities are exactly coassociativity and the two counit identities for \(H\). We use the canonical associativity of ordinary tensors in making these identifications.

The comparison sends \(X\in B\) to \(U(X)\) with coaction

Naturality of \(\eta\) makes every \(U(g)\) a comodule map. Its coassociativity follows from naturality at \(\eta_X\), and its counit equation is the adjunction triangle. We next prove this comparison is an equivalence by constructing its inverse.

4. An actual equalizer inverse on all comodules

Temporarily write an \(H\)-coaction as a \(Q\)-coaction \(\rho:V\to Q(V)\) using (3.2). For it form in \(B\)

The two arrows have common target \(RQ(V)\); the equalizer exists as the kernel of their difference in the abelian category \(B\).

Lemma 4.1. The map \(\rho\) exhibits \(V\) as the equalizer in vector spaces of \(Q(\rho)\) and \(\delta_V\).

Proof. Coassociativity gives the equalizing equation, and \(\varepsilon_V\rho=1_V\) makes \(\rho\) monic. If \(t:W\to Q(V)\) equalizes the pair, put \(s=\varepsilon_Vt\). Naturality of the counit and its comonad identity give

The left inverse \(\varepsilon_V\) makes \(s\) the unique factor. This proves the universal property, not just a split injection. \(\square\)

Since \(U\) is exact, \(U(j_\rho)\) is the equalizer of the same two maps. There are inverse natural vector-space isomorphisms

They follow by comparing the two equalizer universal properties.

This is a comodule isomorphism. To verify the assertion, compare \(Q(\theta_\rho)\rho\) and \(U(\eta_{E_\rho})\theta_\rho\) after \(Q(U(j_\rho))\). Naturality of the unit identifies their composites with \(Q(\rho)\rho\) and \(\delta_V\rho\), respectively. These are equal by the comodule law. The comparison map \(Q(U(j_\rho))\) is monic, since both \(U\) and \(R\) are exact. Cancel it to obtain the required comodule equation.

For a comodule map \(f:(V,\rho)\to(W,\sigma)\), the map \(R(f)j_\rho\) equalizes \(R(\sigma)\) and \(\eta_{R(W)}\). Indeed \(\sigma f=Q(f)\rho\), and naturality of the unit at \(R(f)\) identifies the two resulting composites. It therefore factors uniquely to \(E(f):E_\rho\to E_\sigma\). Uniqueness gives its identity and composition laws; equations (4.3) give naturality of \(\theta\).

Conversely for \(X\in B\), naturality at \(\eta_X\) makes \(\eta_X:X\to RU(X)\) equalize the pair for its comparison coaction \(U(\eta_X)\). It factors to \(s_X:X\to E_{U(\eta_X)}\). Applying \(U\), Lemma 4.1 and the equalizer comparison show that \(U(s_X)\) is invertible. Lemma 2.1 then makes \(s_X\) invertible. The factorization uniqueness proves naturality in \(X\). These \(s_X\) and \(\theta_\rho\) give inverse functor equivalences on all objects and all morphisms:

No separate comonadicity theorem is left as a prerequisite; (4.1) is the inverse just proved.

5. The finite part is exactly the original category

Lemma 5.1. An object \(X\in B\) is isomorphic to a constant object of \(A\) if and only if \(U(X)\) is finite-dimensional.

Proof: subobjects of constants. Let \(Z\subseteq a\), with \(a\in A\). Present \(Z\) as a filtered colimit of constants \(b_j\). Exact filtered colimits in \(B\) make \(Z\) the directed union of the images of the composites \(b_j\to a\). Each composite is a map in \(A\), by full faithfulness of the constant embedding, and its image is a constant object because that embedding is exact.

The images under \(U\) form a directed family of subspaces of the finite-dimensional \(F(a)\). A member of greatest dimension exists. Directedness implies that it contains every member: any common upper member has the same maximal dimension. Faithfulness and exactness of \(U\) reflect the resulting equalities of subobjects. Thus this member is all of \(Z\), proving \(Z\) is constant. Its quotient in \(a\) is constant too, by exactness of the embedding.

Proof: arbitrary finite fibres. Present \(X\) as a filtered colimit of constants \(a_j\). Its image subobjects form a directed union equal to \(X\). The kernel of each \(a_j\to X\) is a subobject of a constant and hence constant by the first part. Consequently its quotient image is also constant. If \(U(X)\) is finite-dimensional, the same greatest-dimension argument shows that one image is all of \(X\). Thus \(X\) is constant. The other implication is the definition of \(F\). \(\square\)

It follows from (4.4) that its restriction is an equivalence \(A\simeq\operatorname{Comod}^{\rm fd}(H)\), over \(F\). In particular all finite comodules occur, not just the images of a chosen family of generators.

The hypotheses also imply that every object of \(A\) has finite length and that every Hom space is finite-dimensional. Strict subobject inclusions give strict inclusions under the exact faithful \(F\), so the dimension strictly increases; quotient chains give the corresponding decreasing bound. These bounds produce a finite composition series by successively taking a maximal proper subobject until zero. Faithfulness embeds \(\operatorname{Hom}_A(a,b)\) into the finite-dimensional space of linear maps \(F(a)\to F(b)\). Neither assertion requires a finite number of isomorphism classes of simple objects.

6. Identify the coefficient quotient with the recovered coalgebra

The equivalence so far produces \(H\) from the right adjoint. We must identify it with the explicit \(C_F\) of (1.1).

For any coalgebra \(H\), and a finite-dimensional right comodule \(M\), write

These coefficients respect every comodule morphism and therefore induce a map from the coefficient quotient of the forgetful functor on finite comodules to \(H\). Formula (1.2) and the coaction equation show it is a coalgebra map.

Proposition 6.1. This coefficient map is invertible.

Proof: surjectivity. Every \(h\in H\), considered in its regular right comodule, lies in a finite-dimensional subcomodule \(M\), by the finite-subcomodule theorem. Take \(\phi=\epsilon_H|_M\). The counit identity gives \(c_M(\phi,h)=h\), so coefficients span all of \(H\).

Proof: injectivity. A finite sum of coefficient classes can be written as one class \([\phi\otimes m]_M\), by taking the direct sum of its finite comodules and using the inclusions to combine the classes. Suppose its coefficient is zero. The map

is a comodule map, by coassociativity. Its image \(E\) is a finite-dimensional subcomodule of the regular comodule, and we may regard \(g\) as a map \(M\to E\). The counit gives \(\epsilon_H|_E\,g=\phi\). Relation (1.1) in this finite-comodule category now yields

Thus every kernel class is zero. \(\square\)

Apply the proposition to the equivalence from Section 5. Coefficient quotients are unchanged by an equivalence preserving the underlying functor up to its specified natural isomorphism: send the coefficient at each object through that isomorphism; full faithfulness carries every relation to a relation, and essential surjectivity represents every object. A quasi-inverse gives the inverse coefficient map, and the natural isomorphisms make the two composites identities by (1.1). Hence \(C_F\simeq H\) as coalgebras.

Under this identification the coaction of each \(F(a)\) is precisely (1.5), because evaluating its coefficients against each basis functional recovers (6.1). Thus the original, explicitly defined \(K_0\) is the finite equivalence. Combining it with (4.4) proves both equivalences in (0.2), with the stated underlying functors. This also proves independence of the small representative category and of the chosen adjoint: the universal coefficient relations provide the canonical compatible coalgebra isomorphisms.

7. Natural operators and the finite topology

Let \(\operatorname{End}(F)\) be the algebra of all natural endomorphisms of \(F\), with composition as multiplication. There are inverse linear identifications

Naturality is exactly relation (1.1), so the displayed functional is well defined. Conversely a functional on \(C_F\) gives, for each \(v\in F(a)\), a functional on \(F(a)^*\). Finite-dimensional biduality gives its unique vector \(t_a(v)\); linearity gives an endomorphism, and (1.1) gives its naturality. These procedures are inverse.

In basis notation \(h_{ij}=[e_i^*\otimes e_j]_a\), formula (1.2) is \(\Delta h_{ij}=\sum_\ell h_{i\ell}\otimes h_{\ell j}\). Hence the matrix of the transformation for \(q\in C_F^*\) is \((q(h_{ij}))\), and multiplication of two such matrices is convolution:

Thus (7.1) is an algebra isomorphism in the stated composition order, and its unit is the counit of \(C_F\).

Give \(\operatorname{End}(F)\) the topology whose basic neighbourhoods of zero are finite intersections of kernels of \(t\mapsto t_a\). These kernels are two-sided ideals. Under (7.1), the kernel for \(a\) annihilates exactly the span of its matrix coefficients, a finite-dimensional subcoalgebra by (1.2).

Conversely every finite-dimensional subcoalgebra \(E\subseteq C_F\) is a finite right comodule, and Section 5 realizes it as some fibre comodule. Its matrix coefficient span is all of \(E\), since applying its restricted counit recovers each element as in Proposition 6.1. Thus these coefficient spans are cofinal among finite-dimensional subcoalgebras. The topology in (7.1) is exactly the finite topology, with open ideals \(E^\perp\), and

Here \(E\) ranges over finite-dimensional subcoalgebras and \({}^\vee\) means continuous linear dual to the discrete field. For the first assertion, a compatible family of restrictions defines a unique functional on their union \(C_F\); finite sums of subcoalgebras give common bounds. For the second, a continuous functional on \(C_F^*\) kills some \(E^\perp\), hence factors through \(E^*\), whose dual is \(E\). Conversely evaluation at any element of \(C_F\) has that continuity. These are inverse identifications.

The complete algebra is not generally finite-dimensional or discrete. By the rational-module equivalence in the comodule prerequisite, \(A\) is also the category of finite-dimensional discrete continuous left modules over \(\operatorname{End}(F)\), with the same underlying vector spaces and all module maps. Continuity is part of this assertion; arbitrary algebraic modules over its underlying algebra give a different category in general.

8. Four graded exercises with complete solutions

Exercise 1 (foundation: ordinary finite vector spaces)

Take \(A=\operatorname{Vect}^{\rm fd}_k\) and \(F\) its identity functor, using a small skeleton. Compute \(C_F\), its coproduct and counit, and all natural operators on \(F\).

Solution. For \(v\in V\), use the map \(\ell_v:k\to V\) in relation (1.1). It gives \([\phi\otimes v]_V=\phi(v)g\), where \(g=[1^*\otimes1]_k\). The counit sends \(g\) to \(1\), so this is a one-dimensional quotient, not a further quotient of it. Formula (1.2) gives \(\Delta g=g\otimes g\) and \(\epsilon g=1\). Hence \(C_F=k\) with its usual one-dimensional coalgebra.

Every natural operator is scalar multiplication: its value on \(k\) is multiplication by a scalar, and naturality at all \(\ell_v\) gives that scalar on every vector. Conversely scalar multiplication is natural. Thus \(\operatorname{End}(F)=k\) with its discrete topology, since its evaluation on \(k\) is faithful. A right \(k\)-coaction has to be \(v\mapsto v\otimes g\) by its counit equation. Therefore finite and arbitrary comodules recover finite and arbitrary vector spaces. The Ind equivalence is the union of finite-dimensional subspaces with the usual linear maps.

Exercise 2 (standard: finite fibres can reconstruct an infinite coalgebra)

Let \(I\) be any infinite small set, and let \(A\) consist of finite-dimensional \(I\)-graded vector spaces, with grade-preserving maps. Let \(F\) forget the grading. Compute the coalgebra and topological operator algebra, and describe all comodules of the reconstructed coalgebra.

Solution. This is an essentially small abelian \(k\)-linear category, with kernels and cokernels in each grade; \(F\) is exact and faithful. Each object has only finitely many nonzero grades. Let \(k_i\) be a one-dimensional object in grade \(i\), and let \(g_i\) be its coefficient class.

The inclusions of the grade summands, and maps \(k_i\to V_i\) taking \(1\) to each vector, show as in Exercise 1 that every coefficient is a linear combination of the \(g_i\). These classes are independent: the rule sending \([\phi\otimes v]_V\) to \(\sum_i\phi(v_i)g_i\) in the abstract direct sum \(\bigoplus_i kg_i\) respects all grade-preserving maps and sends each named class to its distinct basis vector. Thus it is an inverse to the spanning map.

The coproduct and counit are \(\Delta g_i=g_i\otimes g_i\), \(\epsilon g_i=1\). Hence \(C_F=\bigoplus_i kg_i\) as a coalgebra. Equation (7.1) gives \(\operatorname{End}(F)=\prod_i k\), with pointwise multiplication. Its topology is the product of discrete copies of \(k\): killing the operators on finitely many \(k_i\)'s fixes precisely finitely many coordinates, and any finite graded object uses only finitely many coordinates.

For a right comodule, write its finite coefficient expression as \(\rho(m)=\sum_i p_i(m)\otimes g_i\). Coassociativity gives \(p_jp_i=0\) for \(j\ne i\) and \(p_i^2=p_i\), while the counit gives \(m=\sum_i p_i(m)\), with finitely many terms for each vector. Thus \(M=\bigoplus_i\operatorname{im}p_i\), and its maps are exactly grade-preserving maps. Conversely every such direct-sum grading gives that coaction and satisfies both laws. Finite comodules have finite total dimension, so recover exactly \(A\); arbitrary graded spaces give its Ind category. Infinite \(I\) makes the coalgebra infinite-dimensional and its dual topology nondiscrete, despite every original fibre being finite-dimensional.

Exercise 3 (advanced: dual numbers and the actual cofree adjoint)

Let \(D=k[z]/(z^2)\), let \(A\) be finite-dimensional left \(D\)-modules, and let \(F\) forget their actions. Identify \(C_F\) with \(D^*\), calculate its two basis coproducts, and give the coaction corresponding to a square-zero operator. Compute the right adjoint of the extended forgetful functor.

Solution. For a coefficient \([\phi\otimes m]_M\), define its image in \(D^*\) by \(d\mapsto\phi(dm)\). Module-map naturality proves the coefficient relation. Every functional on \(D\) occurs by using the regular module \(D\) and \(m=1\). Conversely the module map \(D\to M\), \(d\mapsto dm\), gives

Thus any kernel coefficient is zero, and finite sums combine by direct sums. This proves the map is an isomorphism. Its coproduct is dual to multiplication in \(D\), as follows by evaluating both factors at two elements and using associativity of the action.

Let \(c_0,c_1\) be dual to \(1,z\). Then

A module with \(z\)-action \(N\) has coaction \(\rho(m)=m\otimes c_0+N(m)\otimes c_1\). Expanding the two iterated coactions shows that their difference has coefficient \(N^2(m)\) at \(c_1\otimes c_1\); all other coefficients agree. Hence its coassociativity is exactly \(N^2=0\), and its counit is identity. Intertwining these coactions is precisely commuting with \(N\).

The Ind category is all \(D\)-modules: each vector's cyclic module is spanned by \(m,Nm\), and finite sums give the directed union of finite-dimensional submodules; the finite-basis and finite-action equations give the filtered Hom comparison. The right adjoint is \(R(V)=\operatorname{Hom}_k(D,V)\), with \((d\cdot h)(r)=h(rd)\). A linear map \(t:M\to V\) corresponds to the module map \(m\mapsto(r\mapsto t(rm))\); evaluation at \(1\) is its inverse. These formulas verify naturality and both inverse identities. Finally \(\operatorname{Hom}_k(D,V)\simeq V\otimes D^*\) because \(D\) is finite-dimensional. Thus the adjoint coalgebra is exactly the one computed above, including its cofree structure on arbitrary \(V\).

Exercise 4 (synthesis: faithfulness cannot replace exactness)

For the same \(D\), let \(s=D/(z)\), and define \(F'(M)=U(M)\oplus\operatorname{Hom}_D(s,M)\) on finite-dimensional modules, where \(U\) forgets the action. Prove that it is a faithful \(k\)-linear finite-fibre functor, but that its canonical coefficient-comodule comparison cannot be an equivalence.

Solution. Both summands are additive and \(k\)-linear on maps, and the first detects every nonzero map. They are finite-dimensional for finite \(M\), so \(F'\) has the asserted properties. Consider the short exact sequence \(0\to s\to D\xrightarrow{q}s\to0\), where \(s\to D\) sends \(1\) to \(z\), and \(q\) is the quotient.

The space \(\operatorname{Hom}_D(s,M)\) identifies with the kernel of the \(z\)-action on \(M\). It is one-dimensional both for \(M=D\), generated by the map with value \(z\), and for \(M=s\). Postcomposition with \(q\) sends the first generator to zero. Therefore \(F'(q)\) is the direct sum of the surjective map \(U(D)\to U(s)\) and a zero map \(k\to k\). Its target is two-dimensional and its image one-dimensional, so it is not surjective. In particular \(F'\) is not exact; the dimensions of the middle and end fibres are also \(3,2,2\).

The coefficient quotient and its canonical comparison still exist, because Section 1 needed only finite-dimensional linear fibres. If that comparison were an equivalence with finite-dimensional comodules, it would be an exact functor between abelian categories: equivalences preserve kernels and cokernels by their universal properties. Composing with the exact underlying-vector functor on comodules would make \(F'\) exact, contradicting the computed \(F'(q)\). Thus the comparison is not an equivalence. This pinpoints the hypothesis that the reconstruction proof needs rather than invalidating the coefficient construction itself.

References

[EGNO] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik, Tensor Categories, Section 1.10, especially Theorem 1.10.1. Freely available author-hosted final version. This gives classical reconstruction context. The Ind extension, tensor-comonad identification, equalizer inverse, finite-part argument and coefficient identification used for the result above are supplied here with the linked course prerequisites.