Let $\mathcal{C}$ be the category of categories and $\mathcal{G}$ the category of metagraphs. Define functors:
$||:\mathcal{C} \rightarrow\mathcal{G}$ such that $|C|=G \Leftrightarrow G$ is the underlying metagraph of $C$
and
$<>:\mathcal{G} \rightarrow \mathcal{C}$ such that $\left
=C \Leftrightarrow C$ is freely generated by $G$.
My question is wether $\left<\right>$ and $\left|\right|$ are adjoints; that is:
Whether the following bijections are natural in $C \in \mathcal{C}$ and $G \in \mathcal{G}$: $\hom_{\mathcal{C}}\left(C,\left
\right) \cong \hom_{\mathcal{G}}\left(G,|C|\right)$