Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 5.1.4.7. Suppose we given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{f} \ar [dr] & & Y \ar [dl] \\ & S & } \]

and a morphism $g: S \rightarrow T$. If $f$ is a covariant equivalence relative to $S$, then it is also a covariant equivalence relative to $T$. This follows from the observation that for every left fibration $q: Z \rightarrow T$, we have canonical isomorphisms

\[ \operatorname{Fun}_{T}(Y, Z) \simeq \operatorname{Fun}_{/S}( Y, S \times _{T} Z) \quad \quad \operatorname{Fun}_{T}(X,Z) \simeq \operatorname{Fun}_{/S}(X, S \times _{T} Z ). \]

Similarly, if $f$ is a contravariant equivalence relative to $S$, then it is also a contravariant equivalence relative to $T$.