Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 7.1.6.7. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $V: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be another functor which is isomorphic to $U$ (as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$). Then an object $Y \in \operatorname{\mathcal{C}}$ is $U$-initial if and only if it is $V$-initial. To prove this, let $Z$ be an object of $\operatorname{\mathcal{C}}$ and let $\alpha : U \rightarrow V$ be an isomorphism of functors, so that we have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(Y), U(Z) ) \ar [d]^{ [\alpha _ Z] \circ } \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( V(Y), V(Z) ) \ar [r]^-{ \circ [ \alpha _ Y ] } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(Y), V(Z) ) } \]

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, where the bottom horizontal and right vertical maps are homotopy equivalences. It follows that the upper horizontal map is a homotopy equivalence if and only if the left vertical map is a homotopy equivalence. Similarly, the object $Y$ is $U$-final if and only if it is $V$-final.