Exponential object in a category of graphs I define a digraph as a set $V$ (vertexes) and a relation $E$ (edges) on $V$.
Morphisms of digraph are functions which preserve $E$.
So we have a category.
Does this category have an exponential object? (and so is cartesian closed, I've already prove it has small products.)
 A: This question is answered  positively with references to many older works on this area in the paper R. Brown, I. Morris, J. Shrimpton and C.D. Wensley, `Graphs of
Morphisms of Graphs',  Electronic Journal of Combinatorics+, A1 of
Volume 15(1), 2008. 1-28, available also here. Actually, in addition to the directed graphs you define there are also the reflexive directed graphs, which have functions $s,t: E \to V$ and a function $\epsilon: V \to E$ such that $s\epsilon=t\epsilon=1_V$. You should particularly notice the paper [15] by Lawvere referenced there. Actually, the categories of directed graphs of either type form a topos, which gives these categories many similarities to the category of sets, but also differences, e.g. their logic is not Boolean. One property of a topos is to have a subobject classifier, and this idea is well worth exploring: for the category of sets the subobject classifier is the set $\{0,1\}$ with the inclusion $\tau: \{1\} \to \{0,1\}$, often called "true".  You can try to work out for yourself what is the subobject classifier for your kind of directed graphs! 
Actually it is standard that if $C$ is any small category, and $Sets$ is the category of sets and functions, then the functor category $(Sets)^C$ also has the structure of topos. So there are many potential applications of these ideas. 
