Kerodon

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Construction 4.6.8.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing objects $X$, $Y$, and $Z$. By virtue of Corollary 4.6.8.5, 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 $\operatorname{\mathcal{C}}$.