Warning 7.3.2.12 (02Z0)

Warning 7.3.2.12. For a general diagram

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dr]^{F_1} \ar@ {<=}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\beta } & \\ \operatorname{\mathcal{K}}\ar [ur]^{\delta } \ar [rr]_{F_0} & & \operatorname{\mathcal{D}}, } \]

we cannot always arrange that there exists a functor $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying the requirements of Proposition 7.3.2.11. However, we can always find a functor $F': \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which satisfies $F'|_{\operatorname{\mathcal{K}}} = F_0$, $F'|_{\operatorname{\mathcal{C}}} = F_1$, and the map

\[ \Delta ^1 \times \operatorname{\mathcal{K}}\simeq \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{K}}} \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}} \]

determines a natural transformation $\beta ': F_0 \rightarrow F_1 \circ \delta $ which is homotopic to $\beta $. To see this, set $M = ( \Delta ^1 \times \operatorname{\mathcal{K}}) {\coprod }_{ (\{ 1\} \times \operatorname{\mathcal{K}}) } \operatorname{\mathcal{C}}$, so that the pair $(\beta , F_1)$ determines a morphism of simplicial sets $f: M \rightarrow \operatorname{\mathcal{D}}$. Proposition 5.2.4.5 supplies a categorical equivalence of simplicial sets $\theta : M \rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, so the induced map

\[ \operatorname{Fun}_{ \operatorname{\mathcal{K}}\coprod \operatorname{\mathcal{C}}/}( \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow {\circ \theta } \operatorname{Fun}_{ \operatorname{\mathcal{K}}\coprod \operatorname{\mathcal{C}}/ }( M, \operatorname{\mathcal{D}}) \]

is an equivalence of $\infty $-categories (Corollary 4.5.4.5). It follows that there exists a functor $F': \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ such that $F'|_{\operatorname{\mathcal{K}}} = F_0$, $F'|_{\operatorname{\mathcal{C}}} = F_1$, and $F' \circ \theta $ is isomorphic to $f$ as an object of the $\infty $-category $\operatorname{Fun}_{ \operatorname{\mathcal{K}}\coprod \operatorname{\mathcal{C}}/ }( M, \operatorname{\mathcal{D}})$. The last requirement is a reformulation of the condition that $\beta ' = F'|_{ \Delta ^1 \times \operatorname{\mathcal{K}}}$ is homotopic to $\beta $.