Remark 11.10.7.9 (Two-out-of-Six). Let $S$ be a simplicial set, and let $f: W_{} \rightarrow X_{}$, $g: X_{} \rightarrow Y_{}$, and $h: Y_{} \rightarrow Z_{}$ be morphisms in the slice category $(\operatorname{Set_{\Delta }})_{/S}$. If $g \circ f$ and $h \circ g$ are covariant equivalences relative to $S$, then $f$, $g$, and $h$ are all covariant equivalences relative to $S$. Similarly, if $g \circ f$ and $h \circ g$ are contravariant equivalences relative to $S$, then $f$, $g$, and $h$ are contravariant equivalences relative to $S$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$