Definition 3.2.1.1. Let $X$ and $Y$ be simplicial sets, and let $K \subseteq X$ be a simplicial subset. We say that morphisms $f_0, f_1: X \rightarrow Y$ are homotopic relative to $K$ if the following conditions are satisfied:
The morphisms $f_0$ and $f_1$ have the same restriction to $K$: that is, there is a morphism $\overline{f}: K \rightarrow Y$ satisfying $f_0 |_{K} = \overline{f} = f_{1}|_{K}$.
The morphisms $f_0$ and $f_1$ belong to the same connected component of the simplicial set $\{ \overline{f} \} \times _{ \operatorname{Fun}(K,Y) } \operatorname{Fun}(X, Y )$.