Remark 4.8.5.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $u: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ having image $\overline{u}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{D}}$. For each integer $n$, the requirement that the map
\[ \pi _{n}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), u) \rightarrow \pi _{n}( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y} ), \overline{u} ) \]
is injective or surjective depends only on the isomorphism class of $u$ (as an object of the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$).