Kerodon

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

Proposition 4.6.6.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories and let $Y$ be an object of $\operatorname{\mathcal{C}}$. Then:

$(1)$

If $F(Y)$ is an initial object of $\operatorname{\mathcal{D}}$, then $Y$ is an initial object of $\operatorname{\mathcal{C}}$.

$(2)$

If $F(Y)$ is a final object of $\operatorname{\mathcal{D}}$, then $Y$ is a final object of $\operatorname{\mathcal{C}}$.

Proof. Let $Z$ be an object of $\operatorname{\mathcal{C}}$. If $F(Y)$ is an initial object of the $\infty $-category $\operatorname{\mathcal{D}}$, then the mapping space $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(Y), F(Z) )$ is a contractible Kan complex. Since $F$ is fully faithful, it follows that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ is also a contractible Kan complex. Allowing $Z$ to vary, we conclude that $Y$ is an initial object of $\operatorname{\mathcal{C}}$. This proves $(1)$; the proof of $(2)$ is similar. $\square$