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Work in ordinary set theory with the axiom of choice. A regular category has finite limits, image factorizations as regular epimorphisms followed by monomorphisms, and pullback-stable regular epimorphisms. A regular functor preserves finite limits and regular epimorphisms.

A capital regular category is one whose well-supported objects are well-pointed objects in a category. Here well-supported means $A\to 1$ is a regular epimorphism; well-pointed means a subobject of $A$ containing every point $1\to A$ must be the whole object. This is the convention of Definition 1.3. It permits objects of proper support with no points. Replacing this condition by global points detecting every isomorphism would change the assertion: a three-element chain is regular but cannot admit a conservative finite-limit-preserving functor to such a category.

Here is a regular-logic proof sketch of capitalization of a small regular category. Use a sort for every object of $\mathcal{C}$, a function symbol for every morphism, and regular theory axioms for composition, finite limiting diagrams and regular-epimorphism lifting. Its set models are precisely regular functors $\mathcal{C}\to \mathbf{Set}$. Empty sorts must be allowed.

The key separation lemma is the regular-logic separation of a proper subobject. For $m:S\hookrightarrow A$ proper, the sequentis not derivable: interpreting a derivation in $\mathcal{C}$ would make the monomorphism $m$ a regular epimorphism, hence an isomorphism. The chase completeness for regular logic supplies a set model in which it fails. Briefly, start with a named antecedent element, adjoin witnesses for the existential axioms, and quotient by required equalities. Every consequent about the original parameter arising at a finite stage has a finite proof; thus a nonderivable consequent remains false. The filtered union is the desired model. This witness construction, rather than an assumption that every sort is inhabited, is what keeps proper supports distinguishable.

Since $\mathcal{C}$ is small, choose a set-indexed family $(M_i{)}_{i\in I}$ of regular functors, one detecting each proper monomorphism, and formIt is regular by coordinatewise finite limits and surjections. It reflects invertibility of monomorphisms. If $P(f)$ is invertible, apply this fact to the image inclusion of $f$ and to the diagonal of its kernel pair. They are invertible, so $f$ is both a regular epimorphism and a monomorphism, hence invertible. Thus $P$ is a conservative functor.

The target is a power of sets as a capital regular category. A well-supported family has every coordinate nonempty, and choice extends any coordinate element to a global tuple. A subobject containing every tuple must consequently contain every coordinate element. Therefore it is capital, giving the required isomorphism-reflecting regular functor. This is the separation aspect of the regular representation results originating in Barr's embedding work; full faithfulness is unnecessary here.

The stronger target of a single set category has a precise restriction: $\mathcal{C}$ admits a conservative regular functor to the Category of sets exactly when it is an almost totally supported regular category. This means every object is well-supported or a strict initial object. Here are necessity and sufficiency.

Suppose $V:\mathcal{C}\to \mathbf{Set}$ is regular and conservative. If $VA$ is nonempty, its support in sets is $1$. Since $V$ preserves images, it sends the support inclusion $\sigma (A)\hookrightarrow 1$ to an isomorphism; conservativity makes $\sigma (A)=1$, so $A$ is well-supported.

If $VA$ is empty, every projection $A\times X\to A$ is sent to the unique bijection between empty sets, hence is invertible. This supplies a morphism $A\to X$ for every $X$. The equalizer of any two such morphisms is also sent to a bijection, hence is invertible, proving uniqueness. Thus $A$ is initial. Every arrow $Y\to A$ forces $VY$ to be empty and is then sent to a bijection; conservativity makes it invertible. So $A$ is strict initial.

Conversely suppose this support condition holds. Use $P$ above and takeProducts preserve finite limits and, by choice, coordinatewise surjections, so $V$ is regular. A well-supported object has all coordinates nonempty. A proper strict initial object has a subterminal image in every coordinate, with at least one empty coordinate: otherwise its arrow to $1$ would be sent by $P$ to an isomorphism, contradicting conservativity.

If $Vf$ is a bijection between well-supported objects, the conservativity of set products on totally supported families makes each $M_i(f)$ bijective: a missing coordinate preimage or a pair of equal-image coordinate elements extends to a tuple witnessing failure of surjectivity or injectivity. Hence $P(f)$ is invertible and so is $f$. An arrow between strict initial objects is already invertible; an arrow from a strict initial object to a well-supported object cannot become a bijection between empty and nonempty sets. An arrow into a strict initial object is invertible by definition. These cases prove conservativity and the conservative regular representation in sets criterion.

In a small regular category $\mathcal{C}$, the regular coverage has one-arrow covers: each regular epimorphism $\alpha :B\twoheadrightarrow A$ is a covering family by itself. Identity maps, composites and pullbacks of such arrows are again covers, so this is a basis for a Grothendieck topology. A presheaf is a sheaf precisely when every compatible section over a covering arrow descends uniquely along that arrow.

It is subcanonical. A section of the representable presheaf $yX=\mathcal{C}(-,X)$ over $B$ is a map $g:B\to X$. Its matching condition is $g{\pi }_1=g{\pi }_2$ on the kernel pair $B{\times }_{A}B$. Since a regular epimorphism is the coequalizer of its kernel pair, there is a unique $\bar{g}:A\to X$ with $\bar{g}\alpha =g$. This is exactly the sheaf condition for $yX$.

For a subfunctor $F^{\prime }\subseteq F$ of a regular-coverage sheaf, define local membership byThis is a subfunctor. If $t:C\to A$ and $\alpha$ witnesses membership, pull back $\alpha$ along $t$; the resulting cover of $C$ witnesses membership of $F(t)x$, because $F^{\prime }$ is stable under restriction.

It is a sheaf as well. Let $\beta :B\twoheadrightarrow A$ cover and let $y\in F^{\prime \prime }(B)$ satisfy the kernel-pair matching condition. The sheaf $F$ gives a unique $x\in F(A)$ with $F(\beta )x=y$. Choose a cover $\alpha :C\twoheadrightarrow B$ with $F(\alpha )y\in F^{\prime }(C)$. The composite $\beta \alpha$ is a cover of $A$ and witnesses $x\in F^{\prime \prime }(A)$. Uniqueness comes from $F$. Thus the inclusion $F^{\prime \prime }\hookrightarrow F$ is a mono between sheaves and is closed for the associated local operator.

Moreover, any closed subobject $S\subseteq F$ containing $F^{\prime }$ is itself a sheaf. If $x\in F^{\prime \prime }(A)$, its restriction along a witnessing cover lies in $S(B)$, and the sheaf condition for $S$ descends it to a section in $S(A)$. Its image in $F(A)$ is $x$ by uniqueness. Hence $F^{\prime \prime }$ is the least closed subobject containing $F^{\prime }$, andThis proves the local membership closure for the regular coverage, with a single regular epimorphism as witness.

Suppose $yA$ is the union, in the sheaf topos, of a family of subsheaves $F_i$. The union in presheaves is the pointwise union $F^{\prime }={\bigcup }_i{F}_i$; its closure is the union in sheaves. Thus $1_A\in F^{\prime \prime }(A)$. A witnessing cover $\alpha :B\twoheadrightarrow A$ belongs to $F_i(B)$ for one index $i$. The two restrictions of $\alpha$ to its kernel pair agree. Since $F_i$ is a subsheaf, this matching section descends in $F_i$ to $1_A\in F_i(A)$. Every map $t:C\to A$ is the restriction of $1_A$ along $t$, so $F_i=yA$. Therefore every representable sheaf is irreducible, even with respect to arbitrary unions. There are no empty covering families in this coverage, and $1_A$ also shows that the representable cannot be initial.

Finally, in the regular syntactic category of a regular theory, the context formula $\varphi$ defines an object $A$. Each formula $\varphi \wedge {\psi }_i$ defines a mono $A_i\hookrightarrow A$, and the associated representable subsheaf is its interpretation inside $yA$ in the generic model. A derivation of the coherent disjunction says that these finitely many subsheaves have union $yA$. Irreducibility gives $y{A}_i=yA$ for some $i$. The Yoneda embedding is full and faithful, since the coverage is subcanonical, and therefore reflects this isomorphism. Hence $A_i\hookrightarrow A$ is invertible in the syntactic category: $\varphi$ entails ${\psi }_i$. ThusThe disjunction is an outer coherent conclusion; the theory and all constituent formulas are regular. This disjunction property of regular theories follows from single-arrow descent, rather than from ordinary first-order compactness.

For a small category that is a regular category, use a sort for each object, an operation for each morphism, and regular theory axioms describing finite limits and regular epimorphisms. Its models are regular functors into the Category of sets. For a proper subobject $m:S\hookrightarrow A$, the sequent asserting $\forall x:A\exists y:S,m(y)=x$ cannot be derivable, since its categorical interpretation would make $m$ a cover and hence invertible. The chase completeness for regular logic supplies a model where it fails. It is essential to allow empty sorts.