Kerodon

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Construction 1.3.5.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we let $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ denote the set of homotopy classes of morphisms from $X$ to $Y$ in $\operatorname{\mathcal{C}}$. For every morphism $f: X \rightarrow Y$, we let $[f]$ denote its equivalence class in $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }(X,Y)$.

It follows from Propositions 1.3.4.2 and 1.3.4.7 that, for every triple of objects $X,Y, Z \in \operatorname{\mathcal{C}}$, there is a unique composition law

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

satisfying the identity $[g] \circ [f] = [h]$ whenever $h: X \rightarrow Z$ is a composition of $f$ and $g$ in the $\infty $-category $\operatorname{\mathcal{C}}$.