I want to show the following statement:
Given a small regular category $\mathcal C$ there exists a topos $\widetilde{\mathcal C}$ and a regular functor $F\colon\mathcal C\to\widetilde{\mathcal{C}}$ such that any regular functor $G\colon\mathcal{C}\to\mathcal E$ from $\mathcal C$ to a cocomplete topos $\mathcal E$ there exists an essentially unique functor $\widetilde G\colon\widetilde{\mathcal C}\to\mathcal E$ which preserves finite limits and all colimits such that $\widetilde G\circ F\simeq G$.
This seems very similar to the universal property of the Yoneda embedding, and since regular functors preserve finite limits and regular epis, my hunch is to define a Grothendieck topology on $\mathcal C$ so that Yoneda followed by sheafification preserves regular epis. I think this can be done as follows: for regular epi $f\colon B\to C$ in $\mathcal C$ coequalizing $A\rightrightarrows B$, we let $X$ be the coequalizer of $y_A\rightrightarrows y_B$ in $\widehat{\mathcal C}$ and choose the least Grothendieck topology $j_f$ such that sheafification sends the unique morphism $X\to y_C$ to an isomorphism. Taking the union of all such Grothendieck topologies over all regular epis we get the least Grothendieck topology $j$ such that the composition $\mathcal C\to\widehat{\mathcal C}\to\mathsf{Sh}(C, j)$ is regular.
From here I'd like to show that if $F\colon\mathcal C\to\mathcal E$ is regular, then it is flat and continuous for $j$, hence factors through $\mathsf{Sh}(C, j)$, but I am not sure how to do that.
- Am I on the right track?
- What would be the next step? (if I missing some important theorem I'd appreciate a reference)