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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 $C$


$<>:\mathcal{G} \rightarrow \mathcal{C}$ such that $\left<G\right>=C \Leftrightarrow C$ 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 $G \in \mathcal{G}$: $$\hom_{\mathcal{C}}\left(C,\left<G\right>\right) \cong \hom_{\mathcal{G}}\left(G,|C|\right)$$

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Why is the order swapped? Usually, with adjoints where one is forgetful, you'd want $\hom_{\mathcal C}(\left<G\right>,C)\cong \hom_{\mathcal G}(G,|C|)$. – Thomas Andrews Nov 22 '12 at 16:58
up vote 1 down vote accepted

Yes, stating the adjunction more precisely in the appropriate direction, as Thomas noted.

Extend the category $\mathcal G\cup \mathcal C$ by graph morphisms $G\to C$, this yields to a category $\mathcal B$, in which every category is coreflected by its underlying graph and every graph is reflected by its free category (because of the freeness), hence $$\hom_{\mathcal G}(G,|C|) \cong \hom_{\mathcal B}(G,C)\cong \hom_{\mathcal C}(\langle G\rangle, C). $$

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