Remark 11.10.7.8. Suppose we 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 either a covariant or contravariant equivalence relative to $S$, then it is a weak homotopy equivalence. This follows by combining Remark 11.10.7.7 with Example 11.10.7.4.