Definition 5.1.3.1. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{f} \ar [dr]_{q_ X} & & Y \ar [dl]^{q_ Y} \\ & S, & } \]

so that we can regard $f$ as a vertex of the simplicial set $\operatorname{Fun}_{/S}(X,Y)$ of Construction 3.1.3.7. We say that $f$ is a *homotopy equivalence relative to $S$* if there exists a morphism $g \in \operatorname{Fun}_{/S}(Y,X)$ such that $g \circ f$ and $\operatorname{id}_{X}$ belong to the same connected component of $\operatorname{Fun}_{/S}(X,X)$, and $f \circ g$ and $\operatorname{id}_{Y}$ belong to the same connected component of $\operatorname{Fun}_{/S}(Y,Y)$. In this case, we will refer to $g$ as a *homotopy inverse of $f$ relative to $S$*.