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.