Corollary 11.10.7.12. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{f} \ar [dr] & & Y \ar [dl] \\ & S, & } \]
where $f$ is a monomorphism. If $q: Z \rightarrow S$ is a left fibration of simplicial sets and $f$ is a covariant equivalence relative to $S$, then the induced map $\theta : \operatorname{Fun}_{/S}(Y,Z) \rightarrow \operatorname{Fun}_{/S}(X,Z)$ is a trivial Kan fibration. Similarly, if $q: Z \rightarrow S$ is a right fibration and $f$ is a contravariant equivalence relative to $S$, then $\theta $ is a trivial Kan fibration.