Definition of hypergraph homomorphism

Question

W.Dörfler and D.A.Waller's paper "A category-theoretical approach to hypergraphs" gives the following definitions:

1. A hypergraph is a triple $H = (V,E,f)$ where $V$ is the set of vertices, $E$ is the set of (hyper)edges and $f \colon E \to \mathcal{P}(V) \smallsetminus \{\emptyset\}$ is the function associating with every edges its edges.

2. A hypergraph homomorphism from $H_1 = (V_1, E_1, f_1)$ to $H_2 = (V_2, E_2, f_2)$ is a pair of functions $h = (h_V \colon V_1 \to V_2, \ h_E \colon E_1 \to E_2)$ such that $h_V(f_1(e)) = f_2(h_E(e))$ for all $e \in E_1$ (by abuse of notation, if $f\colon X \to Y$ and $X' \subseteq X$, then I set $f(X') = \{f(x) \mid x \in X'\}$).

Note that this definition of hypergraphs allows loops (an edge connecting only a vertex with itself) and multiple edges (the same set of vertices can be connected by several edges).

My questions are the following.

1. What happens if we relax the definition of hypergraph homomorphism by requiring only that $h_V(f_1(e)) \subseteq f_2(h_E(e))$ for all $e \in E_1$? Which is the intuitive reason why we do not have a hypergraph homomorphism if $h_V(f_1(e)) \not\supseteq f_2(h_E(e))$ for some $e \in E_1$? Has the relaxed definition any pathological consequence?

2. The authors say that

   In the case of hypergraphs which are graphs [that is, the cardinality of $f(e)$ is $1$ or $2$ for all $e \in E$], we note that the above definition reduces to the usual one of graph homomorphism: $h$ sends vertices to vertices and edges to edges, thus preserving adjacency of vertices.

   Does it hold even in the more relaxed definition of hypergraph homomorphism?

I can't visualize any bad or paradoxical consequence of the relaxed definition of hypergraph homomorphism. On the contrary, it seems to me that only with the relaxed definition it holds that the trivial injection of a sub-hypergraph is always a hypergraph morphism (I assume that a hypergraph $H_1 = (V_1, E_1, f_1)$ is a sub-hypergraph of a hypergraph $H_2 = (V_2, E_2, f_2)$ if $V_1 \subseteq V_2$ and $E_1 \subseteq E_2$ and $f_1(e) = f_2(e) \cap V_1$ for all $e \in E_1$). Moreover, it seems to me that the answer to question 2 is positive.