Corollary 4.6.2.25. Let $f: X \rightarrow Y$ be a morphism of Kan complexes. Then $f$ is fully faithful (when regarded as a functor of $\infty $-categories) if and only if it induces a homotopy equivalence from $X$ to a summand of $Y$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$