In *The Elephant* (A.2.1.9), Johnstone defines the notion of a *coverage* on a category $\mathcal{C}$. Quoting verbatim, a coverage on $\mathcal{C}$ is

a function assigning to each object $A$ of $\mathcal{C}$ a collection $T(A)$ of families $\{f_i\colon A_i\to A\mid i\in I\}$ of morphisms with common codomain $A$ (called $T$-

covering families) such that: If $\{f_i\colon A_i\to A\mid i\in I\}$ is a $T$-covering family and $g\colon B\to A$ is any morphism with codomain $A$, there exists a $T$-covering family $\{h_j\colon B_i\to B\mid j\in J\}$ such that each $gh_j$ factors through some $f_i$.

A category $\mathcal{C}$ equipped with a coverage $T$ defines a site, and in turn a topos $\mathbf{Shv}(\mathcal{C})$ of sheaves.

Let's say that $T$ is a *functorial coverage* if, to each covering family $\{f_i\}$ of $A$ and morphism $g\colon B\to A$ as above, we *assign* a $T$-covering $\{h_j\}$ of $B$, in a way that respects identities and composition in $\mathcal{C}$. Suppose further that all the "collections" $T(A)$ are sets. Then we have defined a presheaf $T\colon\mathcal{C}^{op}\to\mathbf{Set}$.

**Question:** Under what conditions is $T$ a sheaf, and what universal property does it have in the topos $\mathbf{Shv}(\mathcal{C})$?

It would seem there should be some kind of dual relationship between $T$ and the subobject classifier $\Omega\in\mathbf{Shv}(\mathcal{C})$: the former describes (generating) covering sieves and the latter describes closed sieves, which are in some sense orthogonal. Comments on this would be useful as well.