Example 11.10.7.3. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{f} \ar [dr] & & Y \ar [dl] \\ & S. & } \]
If $f$ is a homotopy equivalence relative to $S$ (Definition 11.10.6.1), then it is both a covariant and contravariant equivalence relative to $S$ (see Remark 11.10.6.2).