Example 1.4.2.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. If we refer to a commutative diagram $\sigma :$

\[ \xymatrix { C_{00} \ar [r]^{f} \ar [d]^{g} & C_{01} \ar [d]^{g'} \\ C_{10} \ar [r]^{ f'} & C_{11}, } \]

we implicitly assume that $\sigma $ is a map from the entire simplicial set $\Delta ^1 \times \Delta ^1$ to $\operatorname{\mathcal{C}}$. In other words, we assume that we have specified another morphism $h: C_{00} \rightarrow C_{11}$, which is not indicated in the picture, together with a $2$-simplex $\sigma $ witnessing $h$ as the composition of $f$ and $g'$ and a $2$-simplex $\tau $ witnessing $h$ as the composition of $g$ and $f'$.