By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\}: a\neq b \in V\}$. A *graph homomorphism* between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, w\}\in E(G)$ implies $\{f(v), f(w)\} \in E(H)$.

We say that a graph homomorphism $u:G\to H$ is *universal* if for every graph homomorphism $f:G\to H$ there is $v \in V$ such that $f(v) = u(v)$.

Is the composition of two universal graph homomorphisms universal?