Graph structure on graph homomorphisms

Question

Let $G$ and $H$ be directed graphs. I want to form the following new graph $N$:

• The vertices of $N$ are the graph homomorphisms from $G$ to $H$;
• Given $f,g:G\to H$, there is an edge from $f$ to $g$ if and only if for all vertices $x$ of $G$, there is an edge of $H$ from $f(x)$ to $g(x)$.

(Note that this does not use the edges of $G$ in the construction.)

My question is: is this construction known? If yes, how is it called, and where can I read more about it?

Answer 1

I don't know a name for this construction but it is adjoint to the "Cartesian product" of directed graphs (warning: despite the name, this is not the categorical product in the category of directed graphs). That is, writing $[G,H]$ for your graph of homomorphisms from $G$ to $H$, there is a natural bijection between homomorphisms $G\to [H,K]$ and homomorphisms $G\mathbin{\square}H\to K$ where $\square$ denotes the Cartesian product. This makes the category of directed graphs a closed symmetric monoidal category with respect to the Cartesian product.