Definition 11.10.7.1. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{f} \ar [dr] & & Y \ar [dl] \\ & S. & } \]
We say that $f$ is a covariant equivalence relative to $S$ if, for every left fibration $q: Z \rightarrow S$, precomposition with $f$ induces a bijection $\pi _0( \operatorname{Fun}_{/S}(Y,Z) ) \rightarrow \pi _0( \operatorname{Fun}_{/S}(X,Z) )$. We say that $f$ is a contravariant equivalence relative to $S$ if, for every right fibration $q: Z \rightarrow S$, precomposition with $f$ induces a bijection $\pi _0(\operatorname{Fun}_{/S}(Y,Z)) \rightarrow \pi _0(\operatorname{Fun}_{/S}(X,Z))$.