TITLE: Is $\mathrm{Graph}$ cartesian-closed?
QUESTION [10 upvotes]: Let $\mathrm{Graph}$ be the category of simple, undirected graphs with graph homomorphisms. For any graphs $G, H$ we denote by $\text{Hom}(G, H)$ the set of graph homomorphisms $f:G\to H$. (Note that $\text{Hom}(G, H)$ can be empty, for instance if $\chi(G) > \chi(H)$.)
We can make $\text{Hom}(G, H)$ into a graph by saying that $f, g: G\to H$ form an edge if and only if $\{f(v), g(v)\}\in E(H)$ for all $v\in V(G)$. 
Is $\mathrm{Graph}$ cartesian-closed?

REPLY [5 votes]: For the sake of completeness, the interested reader of this question can find more details in the following paper, which addresses in Section 4 the topic in question: "Categories of graphs are cartesian closed." There, the author considers four different categories of graphs. The paper follows a relatively similar exposition to Tim Campion's answer.
@article{Bumby1986,
  title     = {Categorical constructions in graph theory},
  author    = {Richard T. Bumby and Dana May Latch},
  year      = 1986,
  journal   = {International Journal of Mathematics and Mathematical Sciences},
  url       = {https://www.emis.de/journals/HOA/IJMMS/9/1.pdf}
  volume    = 9,
  number    = 1,
  pages     = {1--16},
  doi       = {10.1155/s0161171286000017}
}