We consider $f$ a homomorphism between graphs $G$ and $H$, that is a function from the vertex set of $G$ to the vertex set of $H$ that preserves edges. Given $u,v \in V(G)$, is there any sort of general bound we can provide to the distance of $f(u)$ and $f(v)$ in $H$?
Namely, is it possible to show that the distance of $f(u)$ and $f(v)$ in $H$ is at most the distance of $u$ and $v$ in $G$? My thinking is that given we have a homomorphism that preserves edges, if this were not true we may have some sort of contradiction. However, I can't seem to formalize and prove the statement
If $u$ and $v$ have distance $n$ then there are vertices $v_0=u, v_1, v_2,\dots, v_n$ in $G$ such that $v_{i-1}$ and $v_i$ are neighbors for $i=1,\dots,n.$ Now we look at $f(v_0),f(v_1),\dots,f(v_n).$ Since $f$ is a homomorphism, $f(v_{i-1})$ and $f(v_i)$ are neighbors in $H$ for $i=1,\dots,n.$ This means that $d(f(u), f(v)) \le k,$ as the distance between two points is defined as the length of the shortest path between them.