$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.2.1.22. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{g} & B \ar [d]^{g'} \\ A' \ar [r]^-{f'} & B', } \]
where $g$ and $g'$ are categorical equivalences. Then $f$ is left cofinal if and only if $f'$ is left cofinal.
Proof.
By virtue of Proposition 7.2.1.21, the morphism $f$ is left cofinal if and only if the composite morphism $g' \circ f$ is left cofinal. Similarly, Proposition 7.2.1.6 guarantees that $f'$ is left cofinal if and only if $f' \circ g$ is left cofinal. We conclude by observing that $g' \circ f = f' \circ g$.
$\square$