When is a functorial coverage a sheaf, and what universal property does it have?

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.

• If $T$ is a Grothendieck topology then $T$ will be a $T$-closed subpresheaf of the presheaf of sieves (a.k.a. the subobject classifier in the presheaf topos). This is more or less what axiom (L) says. – Zhen Lin Jun 25 '16 at 17:55
• Thanks Zhen, but in my case, the coverage itself has a lot in common with a certain sort of sheaf that interests me, so I want to understand it rather than its "closure" as a Grothendieck topology. – David Spivak Jun 25 '16 at 18:03
• If I understand the comment by @ZhenLin correctly, he talks about another subpresheaf of $\Omega_{psh}$ (subobject classifier in presheaves), namely the subpresheaf, usually denoted $J_T\rightarrowtail\Omega_{psh}$ whose value on $A$ is the set of all $T$-covering sieves on $A$ (that is, subpresheaves of the representable presheaf on $A$ generated by a covering family). In a sense, $J_T$ is the canonical functorial coverage associated with any $T$, it defines the same category of sheaves. Thus a natural variant of your question might be when is the presheaf $J_T$ actually a sheaf. – მამუკა ჯიბლაძე Jun 25 '16 at 22:23
• However this version is not so interesting in that the associated sheaf of $J_T$ is the terminal (there is the generic dense mono $1\rightarrowtail J_T$), so if $J_T$ is a sheaf then the topology is trivial. I suspect something similar happens in the general case but cannot figure it out right now. – მამუკა ჯიბლაძე Jun 25 '16 at 22:46
• I wouldn't expect that $T$ could have any universal property in the topos of sheaves, since different coverages on the same category can give rise to the same category of sheaves, and so presumably could different functorial coverages. – Mike Shulman Jun 26 '16 at 3:14