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Notation 4.6.8.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$, $Y$, and $Z$. For every morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the composition law of Construction 4.6.8.9 restricts to a morphism of Kan complexes

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \{ f\} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {\circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z), \]

which is well-defined up to homotopy. Note that this morphism depends only on the homotopy class $[f]$ of the morphism $f$. We will denote this map by $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [f]} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ and refer to it as precomposition with $f$. Similarly, for every morphism $g: Y \rightarrow Z$, the composition law of Remark 4.6.8.10 determines a homotopy class of morphisms $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow { [g] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$, which we will refer to as postcomposition with $g$.