The category of hypergraphs as a topos It seems known that the category of hypergraphs is a topos.
I am looking for any reference here, or just a statement of this in the literature, but can't find anything. There is one paper 

A category-theoretical approach to hypergraphs,
  W. Dörfler and D. A. Waller, ARCHIV DER MATHEMATIK, Volume 34, Number 1, 185-192, DOI:10.1007/BF01224952, 1980

which might contain information about that, but I don't have access to this paper (and it might take some time to get a copy, likely a paper-copy).
By a hypergraph I mean here a triple $(V,E,h)$, where $V$, $E$ are arbitrary sets, while $h$ is a map from $E$ to the set of finite subsets of $V$ (so $V$ is the set of vertices, $E$ the set of hyperedge-labels, and $h$ yields the hyperedge of a hyperedge-label). Morphisms are pairs $a: V \rightarrow V'$, $b: E \rightarrow E'$, which fulfill the usual commutativity condition.
 A: One can reinterpret a hypergraph as a span-shaped diagram of sets where the left leg of the span is a finite map (meaning, all preimages are finite). Indeed, given a hypergraph, consider the span
$$V\leftarrow\lbrace(v,e)\in V\times E\mid v\in h(e)\rbrace\rightarrow E;$$
it is clear that this gives a correspondence.
This seems more natural to work with.
The category of sets is a topos. The category of diagrams of some given shape in a topos is itself a topos, so the category of span-shaped diagrams of this sort is again a topos. Imposing finiteness conditions tends not to destroy the property of being a topos, and one can rapidly check that philosophy in this case.
A: According to this presentation:

Will Grilliette and Lucas Rusnak, Natural Generalizations of Graphs Part II: Commas, Topoi, & Homomorphisms,
  Discrete Mathematics Seminar, Texas State University (2017) DOI:10.13140/RG.2.2.13627.92961

Quote:

The classical categories H of hypergraphs and M of (undirected) multigraphs arise naturally as a comma category using the power-set functor P. However, P is well-known not to preserve limit processes, and both H and M fail to be Cartesian closed as a result, among other issues. On the other hand, the category Q of quivers arises equivalently as both a comma category and a functor category. Consequently, Q can be represented as a topos of presheaves, inheriting a significant amount of structure immediately. 

note: quiver = directed multigraph.
