Definition 7.1.4.1. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that an object $Y \in \operatorname{\mathcal{C}}$ is $U$-final if, for every object $X \in \operatorname{\mathcal{C}}$, the functor $U$ induces a homotopy equivalence
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(X), U(Y) ). \]
We say that $Y$ is $U$-initial if, for every object $Z \in \operatorname{\mathcal{C}}$, the functor $U$ induces a homotopy equivalence
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(Y), U(Z) ). \]