Example 1.5.2.8 (Commutative Squares in a Category). Let $K = \operatorname{\partial }( \Delta ^1 \times \Delta ^1 )$ be as in Example 1.5.2.4. For any ordinary category $\operatorname{\mathcal{C}}$, we can display a diagram $\sigma : K \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ pictorially as
The diagram $\sigma $ is commutative if and only if we have $g' \circ f = f' \circ g$ in $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_{00}, C_{11} )$. In this case, Proposition 1.5.2.6 ensures that $\sigma $ extends uniquely to a diagram $\overline{\sigma }: \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, or equivalently to a functor of ordinary categories $[1] \times [1] \rightarrow \operatorname{\mathcal{C}}$.