Definition 4.6.7.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that an object $Y \in \operatorname{\mathcal{C}}$ is initial if, for every object $Z \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ is a contractible Kan complex. We say that $Y$ is final if, for every object $X \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is a contractible Kan complex.
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