Remark 4.6.9.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$, $Y$, and $Z$. Then the composition law
\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]
of Construction 4.6.9.9 induces a map of sets
\[ \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) ) \times \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) \rightarrow \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) ). \]
Concretely, this map is given by the construction $([g], [f]) \mapsto [h]$, where $h$ is a composition of $f$ and $g$ in the sense of Definition 1.4.4.1.