Example 7.6.3.7 (03GA): Square Diagrams in $\operatorname{\mathcal{QC}}$

Example 7.6.3.7 (Square Diagrams in $\operatorname{\mathcal{QC}}$). Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. Using Remark 7.6.3.6, we see that the data of a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d]^-{F_0} \\ \operatorname{\mathcal{C}}_1 \ar [r]^-{F_1} & \operatorname{\mathcal{C}}} \]

in the $\infty $-category $\operatorname{\mathcal{QC}}$ is equivalent to the data of an $\infty $-category $\operatorname{\mathcal{C}}_{01}$ equipped with functors

\[ G_0: \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_0 \quad \quad G_1: \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_1 \quad \quad H: \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}} \]

together with natural isomorphisms $\alpha _0: (F_0 \circ G_0) \xrightarrow {\sim } H$ and $\alpha _1: (F_1 \circ G_1) \xrightarrow {\sim } H$. In this case, we can identify the data of the tuple $(G_0, \alpha _0, G_1, \alpha _1, H)$ with a single functor of $\infty $-categories

\[ G: \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{ \operatorname{\mathcal{C}}} ( \operatorname{\mathcal{C}}_1 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}). \]