Corollary 10.3.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then $f$ is an isomorphism if and only if it is both a monomorphism and a quotient morphism.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Without loss of generality, we may assume that $f$ is a monomorphism. Applying Example 10.3.3.5, we see that the isomorphism class $[X] \in \operatorname{Sub}(Y)$ is an image of $f$. It follows that $f$ is an isomorphism if and only if $\operatorname{im}(f) = [Y]$. By virtue of Proposition 10.3.3.6, this is equivalent to the requirement that $f$ is a quotient morphism. $\square$