Locally presentable coefficients and accessible sheafification

Written by GPT-6.1 Sol (OpenAI). Original exposition: CC0.

Sheafification exists for every locally presentable coefficient category on a small site. The coefficient category need not have conservative tests preserving ordinary directed colimits. We encode both coefficient objects and covering conditions by a set of unique-extension tests, construct the associated reflection, and identify its actual maps. Limits of sheaves are computed pointwise; sufficiently filtered colimits are too. Finite-limit preservation of the reflection needs an additional hypothesis, as an explicit example with reduced abelian groups shows.

Use Presentable probes and accessible adjoints, Sections 1–4 for presentable objects, the restricted Yoneda image, completeness and raising regular bounds. Small orthogonality and accessible reflective localizations, Sections 1–5 proves the unique-extension reflector and local presentability of a small orthogonality class. Coefficient sheafification on arbitrary sieve sites, Section 1 gives the matching-limit convention. Its directed-test theorem has stronger hypotheses and also proves finite-limit preservation. Finitely presented generators and coherent Ind reconstruction, Sections 1–4 supplies the ordinary module finite-presentation facts used in Exercise 3.

1. Coefficients and the precise sheaf condition

Work in a fixed universe, with choice. Let $\mathcal C$ be a small category with a Grothendieck topology $J$ specified by covering sieves. Fiber products, a terminal object and subcanonicality are not assumed. Let $\mathcal A$ be locally $\kappa$-presentable for an infinite regular cardinal $\kappa$. Thus it is locally small and cocomplete, and every object is a small $\kappa$-filtered colimit of objects in a small family of $\kappa$-presentable objects. An object $b$ is $\kappa$-presentable when its represented Hom preserves those colimits.

Choose a small full skeleton $B$ of the $\kappa$-presentable objects. It admits the $\kappa$-small colimits, with chosen colimit cones transported to its representatives. The prerequisite proves essential smallness and closure: Hom out of such a colimit is a limit of fewer than $\kappa$ Hom sets, which commutes with $\kappa$-filtered colimits. It also proves that $\mathcal A$ has all small limits and that $Y:\mathcal A\longrightarrow \mathsf{Set}^{B^{op}}, \qquad Y(a)(b)=\operatorname{Hom}_{\mathcal A}(b,a) \tag{1.1}$ is fully faithful. Its image consists precisely of presheaves sending every $\kappa$-small colimit cone in $B$ to a limit cone of sets. All these identifications include maps.

For a sieve $S\subseteq h_U=\mathcal C(-,U)$, write $\mathcal E_S$ for its category of arrows to $U$. An object is $f:V\to U$ in $S$; an arrow from $ft:W\to U$ to $f$ is the specified $t:W\to V$. For $F:\mathcal C^{op}\to\mathcal A$, set $M_S(F)=\lim_{(f:V\to U)\in\mathcal E_S^{op}}F(V). \tag{1.2}$ The restriction cone gives $\rho_{S,F}:F(U)\to M_S(F)$. The category $\operatorname{Sh}_J(\mathcal C,\mathcal A)$ is the full subcategory where all these maps, for covering $S$, are invertible.

Applying $\operatorname{Hom}_{\mathcal A}(b,-)$ to (1.2) gives the matching limit of the tested set presheaf, with the same projections. The tests for $b\in B$ therefore detect the sheaf condition, since the fully faithful $Y$ reflects isomorphisms. This agrees with testing by every object of $\mathcal A$: represented Hom preserves limits, and every invertible matching map passes every represented test. Empty covering sieves are included. Their matching object is terminal, so they require $F(U)$ to be terminal.

2. The ambient presheaf category is locally presentable

We will use $\mathcal P=\mathsf{Set}^{E^{op}}$ for a small category $E$. Here is the needed local-presentability proof, including finite relations.

Lemma 2.1. The category $\mathcal P$ is locally finitely presentable.

Proof. Limits and colimits of set presheaves are computed pointwise: take the limits or colimits of sets and define restrictions using their universal properties. Identities, composition and the universal maps hold after each evaluation. Thus all small colimits exist. A representable $h_e$ is finitely presentable, because $\operatorname{Hom}(h_e,X)=X(e)$ naturally and evaluation preserves filtered colimits.

For $X\in\mathcal P$, introduce a generator $g=(e,x)$ for every $x\in X(e)$, with its evaluation $h_e\to X$. Their coproduct is pointwise surjective: $x\in X(e)$ is the image of $1_e$ in its own summand. For each $v:d\to e$ and $x\in X(e)$, impose the relation between $h_d\xrightarrow{\ 1\ }h_d \xrightarrow{\ j_{(d,X(v)x)}\ }\coprod_g h_{\operatorname{ob}g} \quad\text{and}\quad h_d\xrightarrow{\ h_v\ }h_e \xrightarrow{\ j_{(e,x)}\ }\coprod_g h_{\operatorname{ob}g}. \tag{2.1}$ These are a small set of relations. A map from their coequalizer to $Z$ is a family of elements $z_{e,x}\in Z(e)$ satisfying $Z(v)z_{e,x}=z_{d,X(v)x}$. Such families are exactly the natural transformations $X\to Z$, with inverse evaluation at each $x$. The coequalizer is therefore $X$, with the displayed evaluation maps.

Let $(G,T)$ range over finite sets of generators and relations such that both endpoint generators of every relation in $T$ belong to $G$. The corresponding finite coequalizer is $Q_{G,T}= \operatorname{Coeq}\left( \coprod_{r\in T}h_{\operatorname{dom}r} \rightrightarrows \coprod_{g\in G}h_{\operatorname{ob}g}\right). \tag{2.2}$ The empty pair is allowed; finite unions give common upper bounds. Inclusions induce the unique coequalizer maps and obey both functor laws. Each $Q_{G,T}$ is finitely presentable: Hom out of (2.2) is the equalizer of two maps between finite products of evaluation sets, and filtered colimits of sets commute with these finite limits.

The canonical maps give $\pi:\operatorname{colim}_{G,T}Q_{G,T}\to X$. Conversely the singleton generator pairs give a map from the full generator coproduct to this colimit. Each relation holds at a finite pair containing its two endpoints and that relation. Hence this map factors through $X$. Both composites are identities on every generator; the coproduct and colimit universal properties make them identities. This proves the filtered presentation of $X$, including its actual structural maps. Up to isomorphism there is a small set of objects (2.2): finite lists of objects and arrows of the small $E$ specify them. This is the required small family of finitely presentable objects. $\square$

3. Encode coefficient continuity and covering sieves

Put $E=\mathcal C\times B$, so $\mathcal P=\mathsf{Set}^{(\mathcal C\times B)^{op}}, \qquad \widetilde F(U,b)=\operatorname{Hom}_{\mathcal A}(b,F(U)). \tag{3.1}$ Currying (1.1) gives a fully faithful functor $\widetilde{(-)}:\operatorname{Fun}(\mathcal C^{op},\mathcal A)\to\mathcal P$. We specify its image by actual maps in $\mathcal P$.

For $U\in\mathcal C$ and a chosen $\kappa$-small colimit cone $(b_i\to b)_{i\in I}$ in $B$, form $t_{U,(b_i)}: \operatorname{colim}_{i\in I}h_{(U,b_i)} \longrightarrow h_{(U,b)}. \tag{3.2}$ Let $T_0$ be the set of these maps. It is a set: choose $\kappa$-small category structures on ordinals below $\kappa$, then all their diagrams in the small $B$, their chosen cones, and all $U$. Relabelling index categories changes none of the required conditions.

Recall that $X\perp s:A\to D$ means restriction $\operatorname{Hom}(D,X)\to\operatorname{Hom}(A,X)$ is bijective. Yoneda and the colimit universal property identify orthogonality to (3.2) with the specified comparison $X(U,b)\longrightarrow\lim_{i\in I^{op}}X(U,b_i). \tag{3.3}$ It follows from (1.1) that $\operatorname{Fun}(\mathcal C^{op},\mathcal A) \simeq T_0^\perp\subseteq\mathcal P. \tag{3.4}$ To check the functorial assertion, recover $F(U)$ from each continuous row $X(U,-)$. A restriction between rows lifts uniquely through the full $Y$. Uniqueness proves the identity and composition laws. A transformation of such $X$'s lifts uniquely at each $U$, and its naturality follows by the same fullness test. Thus (3.4) recovers all arrows, not just objects.

For a covering sieve $S$ on $U$ and $b\in B$, the arrows $f:V\to U$ give the map $t_{S,b}: \operatorname{colim}_{(f:V\to U)\in\mathcal E_S}h_{(V,b)} \longrightarrow h_{(U,b)}. \tag{3.5}$ The diagram sends an arrow specified by $t:W\to V$ to $h_{(W,b)}\to h_{(V,b)}$; its map to $h_{(U,b)}$ is induced by $f$. This fixes the direction of the indexing category. There is a set $T_1$ of all maps (3.5): covering sieves are subsets of the arrow sets of the small site.

For $X=\widetilde F$, restriction along (3.5) is exactly $\operatorname{Hom}_{\mathcal A}(b,F(U)) \longrightarrow \lim_{\mathcal E_S^{op}}\operatorname{Hom}_{\mathcal A}(b,F(V)). \tag{3.6}$ The limit on the right is $\operatorname{Hom}_{\mathcal A}(b,M_S(F))$, with its original projections. Section 1 shows that all maps (3.6) are bijective precisely when $F$ is a sheaf. Consequently $\operatorname{Sh}_J(\mathcal C,\mathcal A) \simeq (T_0\cup T_1)^\perp\subseteq\mathcal P. \tag{3.7}$ For an empty covering sieve, (3.5) has initial domain, so its test requires $X(U,b)$ to be a singleton. This is exactly the terminal-value condition already identified; no nonempty-cover assumption has entered.

4. The reflector, its maps and its colimits

Theorem 4.1. The inclusion $j:\operatorname{Sh}_J(\mathcal C,\mathcal A) \longrightarrow\operatorname{Fun}(\mathcal C^{op},\mathcal A) \tag{4.1}$ has an accessible left adjoint $a$. The sheaf category is locally presentable. Its small limits are computed pointwise. For a sufficiently large regular $\mu$, its $\mu$-filtered colimits are computed pointwise as well.

Proof of the adjunction. Lemma 2.1 and the small orthogonality theorem give a reflector $L:\mathcal P\longrightarrow (T_0\cup T_1)^\perp. \tag{4.2}$ The cited theorem constructs it by simultaneous pushouts, coequalizers imposing uniqueness and a regular transfinite chain. Every endpoint in (3.2) and (3.5) has a common presentability bound, as checked below. Thus it supplies the reflector, not merely a weak reflection.

Apply (4.2) to $\widetilde F$, then use (3.7) to recover $aF$. If $G$ is a sheaf, the actual restriction maps give $\begin{aligned} \operatorname{Hom}_{\mathrm{Sh}}(aF,G) &\simeq\operatorname{Hom}_{\mathcal P}(L\widetilde F,\widetilde G)\\ &\simeq\operatorname{Hom}_{\mathcal P}(\widetilde F,\widetilde G)\\ &\simeq\operatorname{Nat}(F,jG). \end{aligned} \tag{4.3}$ The middle bijection restricts along the orthogonality unit; the other two are the fully faithful coefficient identifications. Their inverses respectively lift rows and extend uniquely from that unit. Both inverse identities and both-variable naturality follow from those same identities for the three bijections.

In particular the unit $\eta_F:F\to jaF$ is the unique coefficient transformation whose image is the unit $\widetilde F\to L\widetilde F$. This determines its components and their naturality. For a sheaf $G$, the identity on $jG$ extends uniquely to an invertible counit $\varepsilon_G:ajG\to G$. The two composites in the triangle identities restrict to the identity on the corresponding unit, so uniqueness makes them identities. Thus $ja$ is an idempotent monad with multiplication $j\varepsilon_a$. These statements concern the specified maps, and do not assume pointwise choices of unrelated reflections.

Limits and colimits. Orthogonality classes are closed under all small limits: represented Hom turns a limit of their tests into a limit of bijections. Moreover (3.1) preserves pointwise coefficient limits, since each represented Hom preserves limits. Therefore a pointwise limit of coefficient sheaves is a sheaf with its original projections. All small limits exist in $\mathcal A$, so these are the desired limits.

For a small diagram $(G_i)$ of sheaves, put $H=\operatorname{colim}_i jG_i$, computed pointwise in $\mathcal A$. Its sheaf colimit is $aH$, with legs $G_i\xrightarrow{\ \varepsilon_{G_i}^{-1}\ }ajG_i \xrightarrow{\ a(c_i)\ }aH. \tag{4.4}$ Indeed (4.3) turns a map from $aH$ to a sheaf into precisely a compatible family from the $G_i$. The stated legs give that family, which proves existence and uniqueness with the actual maps.

The cardinal bound. Choose an infinite regular $\mu\ge\kappa$ with $|\operatorname{Mor}\mathcal C|<\mu$. A representable in $\mathcal P$ is $\mu$-presentable. Each domain of (3.2) is a colimit of fewer than $\kappa$ representables. Each category $\mathcal E_S$ has fewer than $\mu$ objects and arrows: its objects are site arrows, and its arrows are specified by composable pairs of site arrows. The domain of (3.5) is consequently a colimit of a $\mu$-small diagram of representables. Hom out of either domain is a $\mu$-small limit of evaluation functors, so the $\mu$-filtered Set comparison proves its $\mu$-presentability. The codomains are representables too.

Lemma 2.1 and the cardinal-raising theorem make $\mathcal P$ locally $\mu$-presentable. Applying the orthogonality theorem at this common endpoint bound proves that the class in (3.7) is locally $\mu$-presentable and that its inclusion in $\mathcal P$ creates $\mu$-filtered colimits. Applying it to $T_0$ alone also proves local $\mu$-presentability of coefficient presheaves. The coefficient embedding (3.1) preserves such pointwise colimits: every $b\in B$ is $\kappa$-presentable, hence $\mu$-presentable. Full faithfulness therefore identifies the created colimit with the pointwise coefficient one. This proves the assertion for (4.1).

Equivalently, apply $\operatorname{Hom}_{\mathcal A}(b,-)$ to the pointwise colimit of a $\mu$-filtered sheaf diagram. Its sheaf comparison is the Set exchange between that colimit and the $\mu$-small matching limit. This comparison is bijective on every $b$, hence invertible in $\mathcal A$, and respects the original projections.

Finally $a$ preserves all colimits as a left adjoint. On a $\mu$-filtered diagram, $j$ creates the resulting sheaf colimit, so the endofunctor $ja$ preserves that pointwise colimit. This is accessibility of the actual reflection as a functor on coefficient presheaves. $\square$

For an empty site category, $E$ and both test families are empty and all three presheaf/sheaf categories in (3.4) and (3.7) are terminal categories; the reflection is the identity on that category. For a terminal coefficient category, every coefficient presheaf and matching cone is unique. The theorem includes these cases.

5. What the stronger test hypothesis adds

If $\mathcal A$ has the conservative tests preserving small limits and ordinary directed colimits assumed in the earlier coefficient theorem, its two-plus construction is another left adjoint to the same inclusion. The unique adjunction comparison is therefore a natural isomorphism between that construction and $a$, compatible with both units. In that setting the earlier proof also gives finite-limit preservation.

Local presentability alone supplies represented tests preserving a sufficiently large filtered class. It does not supply the directed-colimit comparison needed for the two-plus proof. Theorem 4.1 consequently asserts neither that two plus steps always suffice nor that $a$ preserves finite limits. Exercise 4 proves that the latter assertion would be false.

As an example of iterating coefficient categories, let $X$ be a topological space in the working universe. Set sheaves on its small category of opens are locally presentable by Theorem 4.1 with $\mathcal A=\mathsf{Set}$. That sheaf category can itself be used as $\mathcal A$ in Theorem 4.1 on a different small site. The construction then treats sheaves whose values are sheaves on $X$, with actual coefficient morphisms; it does not require section evaluation on every open of $X$ to preserve ordinary directed colimits.

6. Four graded exercises with complete solutions

Exercise 1 (foundation: one covering arrow)

Let the site have objects $u,v$, the sole nonidentity arrow $v\to u$, and the topology generated by that singleton covering of $u$; at $v$ only the maximal sieve covers. For a locally presentable $\mathcal A$, describe every coefficient sheaf and its reflection. Give the unit and both inverse adjunction factors.

Solution. A coefficient presheaf is an arrow $r:F(u)\to F(v)$. The nonmaximal covering sieve at $u$ has one object $v\to u$ and identity arrow. Its matching value is $F(v)$, so the sheaf condition is exactly invertibility of $r$. A sheaf is thus canonically isomorphic to the constant arrow on its value at $v$.

Set $aF=(F(v)\xrightarrow{1}F(v))$, and let its unit have components $r$ at $u$ and $1$ at $v$. For a sheaf $G=(G(u)\xrightarrow{s}G(v))$, a map $f:F\to G$ has components satisfying $sf_u=f_vr$. It extends to the unique map $g:aF\to G$ with $g_v=f_v$, $g_u=s^{-1}f_v$. Conversely $g$ restricts along the stated unit to $f_u=g_ur$, $f_v=g_v$. The displayed equation and invertibility of $s$ verify both inverse identities. Composing a map of source arrows or target sheaf arrows commutes with these component formulas, proving both naturalities. The counit has components $s^{-1}$ and $1$, so the triangle identities follow componentwise. All indexing and matching conditions, including the pullback of the covering sieve to $v$, have been included. $\square$

Exercise 2 (standard: a disjoint cover and an empty cover)

Take the category of opens of a discrete two-point space: $\varnothing,u,v,x=u\cup v$. With its ordinary covers and coefficients in $\mathcal A$, compute $aF$, all its restriction maps, the unit and the corresponding adjunction.

Solution. The empty cover forces a sheaf value at $\varnothing$ to be $1_{\mathcal A}$. The cover $x=u\cup v$, with intersection $\varnothing$, then requires the value at $x$ to be the product of its two values. Thus a sheaf is determined by an arbitrary pair $(A,B)$, with values $A,B,A\times B,1_{\mathcal A}$ at $u,v,x,\varnothing$, respectively. Its restrictions are the two projections and the unique arrows to the terminal object. They satisfy every functor law and all cover matching conditions.

For a presheaf $F$, choose $A=F(u)$, $B=F(v)$. Use these four values for $aF$. The unit is the identity at $u,v$, the unique map at $\varnothing$, and the pair of original restrictions $F(x)\to F(u)\times F(v)$ at $x$. Its naturality at the arrows to $\varnothing$ follows from terminal uniqueness; at the two other generating restrictions it is the product equation.

A transformation to the sheaf determined by $(A',B')$ is uniquely determined by maps $F(u)\to A'$, $F(v)\to B'$: at $x$ use their pair after the restrictions, and at $\varnothing$ use the unique map. These same two maps determine a unique sheaf transformation $aF\to(A',B')$. Restricting along the unit leaves both component maps unchanged, giving inverse natural Hom bijections. This also verifies the unit and counit triangles. Arbitrary presheaf colimits need not already be sheaves: for $\mathcal A=\mathsf{Set}$, the pointwise initial presheaf has empty value at $\varnothing$, whereas the initial sheaf has singleton value there. $\square$

Exercise 3 (advanced: a locally presentable category without finitary tests)

Let $\mathcal R$ be the full subcategory of abelian groups satisfying $\operatorname{Hom}_{\mathbb Z}(\mathbb Q,M)=0$. Prove that it is locally presentable, that its limits are the ordinary group limits, and that ordinary directed colimits in $\mathcal R$ do not preserve kernels. Conclude that the earlier conservative directed-test hypothesis cannot hold in $\mathcal R$.

Solution. The ordinary module finite-presentation theorem makes $\mathsf{Ab}$ locally finitely presentable: it is cocomplete and is generated under filtered colimits by its small category of finitely presented groups. Apply the small orthogonality theorem to the single map $\mathbb Q\to0$ in $\mathsf{Ab}$. Orthogonality means that the singleton $\operatorname{Hom}(0,M)$ maps bijectively to $\operatorname{Hom}(\mathbb Q,M)$, exactly the defining condition. Hence $\mathcal R$ is a reflective locally presentable category. Write $r:\mathsf{Ab}\to\mathcal R$ for its reflector. Its limits are inherited, because both represented Hom tests preserve limits. Its colimits are the reflections of ordinary group colimits, by the reflection adjunction.

The group $\mathbb Z$ belongs to $\mathcal R$. A homomorphism $f:\mathbb Q\to\mathbb Z$ has $f(1)$ divisible by every positive integer, so $f(1)=0$; torsion-freeness then makes $f(a/b)=0$. Every group killed by a positive integer $m$ is also in $\mathcal R$: $f(q)=mf(q/m)=0$. Moreover $r\mathbb Q=0,\qquad r(\mathbb Q/\mathbb Z)=0. \tag{6.1}$ For the first assertion, all maps from $\mathbb Q$ to an object of $\mathcal R$ are zero by definition, which is the universal property of the zero reflection. A map from $\mathbb Q/\mathbb Z$ to such an object is zero after the surjection from $\mathbb Q$, hence zero; this proves the second assertion by the same universal property.

For $n\ge1$, use the group map $q_n:\mathbb Z\to\mathbb Z/n!\mathbb Z$. Its kernel is $n!\mathbb Z$, identified with $\mathbb Z$ by multiplication by $n!$. Make a directed diagram of these kernel sequences by using the identity on the identified left $\mathbb Z$, multiplication by $n+1$ on the middle $\mathbb Z$, and multiplication by $n+1$ on the quotient. The squares commute because $(n+1)n!=(n+1)!$. Every term belongs to $\mathcal R$.

The middle ordinary group colimit is $\mathbb Q$, with maps $a\mapsto a/n!$. These maps are compatible, every rational appears, and equality of two representatives is witnessed after a common factorial denominator. The quotient colimit is $\mathbb Q/\mathbb Z$, with maps $\bar a\mapsto a/n!+\mathbb Z$, by the same representative and equality tests. The kernel colimit is the constant group $\mathbb Z$. Reflecting these colimits and using (6.1) gives the kernel comparison in $\mathcal R$ $\operatorname{colim}_n\ker q_n=\mathbb Z \longrightarrow \ker\!\left(\operatorname{colim}_n\mathbb Z \longrightarrow\operatorname{colim}_n\mathbb Z/n!\mathbb Z\right) =\ker(0\to0)=0. \tag{6.2}$ It is not invertible.

If a jointly conservative family to sets preserved small limits and ordinary directed colimits, applying it to this comparison would give the bijective filtered Set kernel comparison. Conservativity would make (6.2) invertible, a contradiction. Thus no such family exists in $\mathcal R$. This example satisfies Theorem 4.1 but not the earlier directed-test assumptions. $\square$

Exercise 4 (synthesis: sheafification need not preserve finite limits)

Let $\mathcal C$ have objects $n\in\mathbb N$ and one arrow $m\to n$ exactly when $m\ge n$. Declare every nonempty sieve covering. With coefficients in $\mathcal R$ from Exercise 3, compute the sheaf reflector and exhibit its failure to preserve a kernel.

Solution. A nonempty sieve on $n$ is precisely the set of arrows from $k\ge m$ for some $m\ge n$: upward closure gives this tail, and the least member supplies $m$. Pullback to $l\ge n$ gives the tail starting at $\max(l,m)$, so it is nonempty. The maximal sieve is a tail, and local character holds: if a sieve pulls back to nonempty sieves on every member of one nonempty tail, then it contains an arrow and is therefore itself covering. This is a Grothendieck topology on the small category, with no empty covering sieve.

A presheaf is a direct sequence $F(0)\to F(1)\to\cdots$. The arrow category of the tail starting at $m$ has terminal object $m\to n$. Its opposite has initial object $m$, and the matching limit is $F(m)$: a cone is determined by its component there, with all later components obtained by the sequence maps. Thus the sheaf condition is precisely that every transition $F(n)\to F(m)$ be invertible.

The sheaf category is equivalent to $\mathcal R$ by constant sequences. For a sheaf $G$, the transition maps $G(0)\to G(n)$ form a natural isomorphism from its constant sequence; its inverse uses their inverses. The same equations recover every sheaf transformation from its component at zero.

For an arbitrary sequence, its reflection is the constant sequence on $\operatorname{colim}_{\mathcal R}F(n)$, with the original colimit legs as unit. Indeed a map to a constant sequence is exactly a colimit cocone, with inverse given by its unique factor through that colimit. Composing with the preceding sheaf isomorphism gives the full sheaf adjunction, with both naturalities and its actual unit.

Now use the two sequences and the natural transformation $q_n$ from Exercise 3, with an initial index shift if necessary. Their kernel sequence is constant $\mathbb Z$. The reflected middle and quotient sequences are both constant zero, while the reflected kernel is constant $\mathbb Z$. Their canonical kernel comparison is therefore the noninvertible map (6.2), at every site object. Since a kernel is the equalizer of $q$ and zero, this is a failure to preserve a finite limit. It does not conflict with pointwise sheaf limits or sufficiently filtered pointwise colimits in Theorem 4.1: the ordinary sequence colimit here need not be in that larger filtered class. $\square$

References

The Stacks Project, Definition 7.7.6, defines coefficient-valued sheaves by all represented set-valued tests. Jiří Adámek, Michel Hébert and Lurdes Sousa, The Orthogonal Subcategory Problem and the Small Object Argument, introduction, discusses orthogonality and reflection. The small-set reflector and presentability theorems used here have complete proofs in the linked course prerequisites. The two explicit test families, coefficient adjunction maps and counterexamples are proved above.