Remark 6.2.1.12. Let $F: X \rightarrow Y$ be a morphism of Kan complexes. Then $F$ admits a right adjoint (in the sense of Notation 6.2.1.3) if and only if $F$ is a homotopy equivalence. This follows by combining Remark 6.2.1.10 with Example 6.2.1.11.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$