Kerodon

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

Construction 4.6.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$, $Y$, and $Z$. By virtue of Corollary 4.6.3.5, the restriction map

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]

is a trivial Kan fibration, so its homotopy class $[\theta ]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. We let

\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]

denote the morphism in $\mathrm{h} \mathit{\operatorname{Kan}}$ obtained by composing $[\theta ]^{-1}$ with (the homotopy class of) the restriction map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$. We will refer to $\circ $ as the composition law on the $\infty $-category $\operatorname{\mathcal{C}}$.