Hypergraphs as a category

Question

If we regard hypergraphs $H=(V,E)$ as the objects, where $V$ is a set and $E\subseteq {\cal P}(V)$, what is the "right", or commonly used notion of morphism for that category?

Answer 1

Your question is answered in the Wikipedia article you linked to:

The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms.

A hypergraph homomorphism is a map from the vertex set of one hypergraph to another such that each edge maps to one other edge.

To see that this defines a category, note that identity and associativity of vertex set maps imply identity and associativity of the corresponding hypergraph morphisms, and the edge mapping condition is clearly preserved under composition.