Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 1.4.6.2. Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be a composable pair of morphisms in an $\infty $-category $\operatorname{\mathcal{C}}$, so that the tuple $(g, \bullet , f)$ determines a map of simplicial sets $\Lambda ^2_1 \rightarrow \operatorname{\mathcal{C}}$ (see Exercise 1.1.2.14). Then the fiber product

\[ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \Lambda ^2_1, \operatorname{\mathcal{C}}) } \{ (g, \bullet , f) \} \]

is a contractible Kan complex.