Remark 11.10.6.2. Suppose we are given a commutative diagram of simplicial sets
The following conditions are equivalent:
The morphism $f$ is a homotopy equivalence relative to $S$ (Definition 11.10.6.1).
For every object $B$ of the slice category $(\operatorname{Set_{\Delta }})_{/S}$, postcomposition with $f$ induces a homotopy equivalence of simplicial sets $\operatorname{Fun}_{/S}(B,X) \rightarrow \operatorname{Fun}_{/S}(B,Y)$.
For every object $B$ of the slice category $(\operatorname{Set_{\Delta }})_{/S}$, postcomposition with $f$ induces a bijection $\pi _0( \operatorname{Fun}_{/S}(B,X) ) \rightarrow \pi _0( \operatorname{Fun}_{/S}(B,Y) )$.
For every object $Z$ of the slice category $(\operatorname{Set_{\Delta }})_{/S}$, precomposition with $f$ induces a homotopy equivalence of simplicial sets $\operatorname{Fun}_{/S}(Y,Z) \rightarrow \operatorname{Fun}_{/S}(X,Z)$.
For every object $Z$ of the slice category $(\operatorname{Set_{\Delta }})_{/S}$, precomposition with $f$ induces a bijection $\pi _0( \operatorname{Fun}_{/S}(Y,Z) )\rightarrow \pi _0( \operatorname{Fun}_{/S}(X,Z) )$.