Remark 11.10.7.7. Suppose we are 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 $T$.