Definition 9.3.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. We say that a morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is $n$-truncated if, for every object $C \in \operatorname{\mathcal{C}}$, composition with the homotopy class $[f]$ induces an $n$-truncated morphism of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \xrightarrow { [f] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)$.
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