From Steve Awodey's Category Theory:
We begin with the free category $C(G)$ on some finite graph $G$, and then consider a finite set $\Sigma$ of relations of the form $g_{1}\circ\cdots\circ g_{n} = g'_{1}\circ\cdots\circ g'_{m}$ with all $g_{i}\in G$ and $\mathsf{dom}(g_{n}) = \mathsf{dom}(g'_{m})$ and $\mathsf{cod}(g_{1}) = \mathsf{cod}(g'_{1})$. Such a relation identifies two "paths" in $C(G)$ with the same "endpoints" and "direction." Next, let $\sim _{\Sigma}$ be the smallest congruence $\sim$ on $C$ such that $g \sim g'$ for each equation $g = g'$ in $\Sigma$. Taking the quotient by this congruence, we have a notion of a finitely presented category: $C(G, \Sigma) = C(G)/{\sim_\Sigma}$. Specifically, in $C(G, \Sigma)$ there is a "diagram of type $G$" that is, a graph homomorphism $i : G \to |C(G, \Sigma)|$, satisfying all the conditions $i(g) = i(g')$, for all $g = g' \in \Sigma$. Moreover, given any category $D$ with a diagram of type $G$, say $h : G \to |D|$, that satisfies all the conditions $h(g) = h(g')$, for all $g = g' \in \Sigma$, there is a unique functor $\bar{h} : C(G, \Sigma) \to D$ with $|\bar{h}| \circ i = h$.
- How to prove this UMP?
- "a graph homomorphism $i:G \to |C(G, \Sigma)|$, satisfying all the conditions $i(g)=i(g')$, for all $g=g'\in\Sigma$" But $g = g_{1}\circ\cdots\circ g_{n}$ with all $g_{i}\in G$,and $g \notin G$ in general, so what does $i(g)$ mean?