Corollary 8.1.4.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ admit pushouts. The following conditions are equivalent:
- $(1)$
The functor $F$ carries pushout squares in $\operatorname{\mathcal{C}}$ to pushout squares in $\operatorname{\mathcal{D}}$.
- $(2)$
The induced map $\operatorname{Cospan}(F): \operatorname{Cospan}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{D}})$ is a functor of $(\infty ,2)$-categories, in the sense of Definition 5.4.7.1.
Proof.
The implication $(1) \Rightarrow (2)$ follows immediately from the criterion of Proposition 8.1.4.2. For the converse implication, suppose that $\operatorname{Cospan}(F): \operatorname{Cospan}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{D}})$ is a functor of $2$-categories, and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a pushout square, which we display as a diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] & X_{01}. } \]
Let $\rho : \operatorname{Tw}( \Delta ^2 ) \rightarrow \Delta ^1 \times \Delta ^1$ denote the morphism of simplicial sets given on vertices by the formula $\rho (i,j) = (\max (0, 1-i) , \max (0,j-1) )$. Then $\sigma \circ \rho $ can be identified with a $2$-simplex $\tau $ of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, corresponding to a diagram
\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar [dr]_{\operatorname{id}} & & X \ar [dl] \ar [dr] & & X_1 \ar [dl]^{\operatorname{id}} \\ & X_0 \ar [dr] & & X_1 \ar [dl] \\ & & X_{01} & & } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$. It follows from the criterion of Proposition 8.1.4.2 that $\tau $ is a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. If $\operatorname{Cospan}(F)$ is a functor of $(\infty ,2)$-categories, then it carries $\tau $ to a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{D}})$. Applying the criterion of Proposition 8.1.4.2 again, we conclude that $F(\sigma )$ is a pushout square in $\operatorname{\mathcal{D}}$.
$\square$