Let $G,H$ be simple undirected graphs without loops such that there are graph homomorphisms $f_1:G\to H$ and $f_2: H\to G$.

An easy argument shows that we have $\chi(G) = \chi(H)$, even for graphs with infinite chromatic number.

What other conclusions can be made about $G, H$ if we know that there are graph homomorphisms in either direction?

**Note**: There need not be a graph homomorphism between two graphs. For instance, $\textrm{Hom}(G,K_2)=\emptyset$ iff $\chi(G) > 2$.