Proposition 3.2.1.4 (00V9)

Proposition 3.2.1.4. Let $f,g: X \rightarrow Y$ be morphisms of simplicial sets and let $K \subseteq X$ be a simplicial subset. Then:

The morphisms $f$ and $g$ are homotopic relative to $K$ if and only if there exists a sequence of morphisms $f = f_0, f_1, \ldots , f_ n = g$ from $X_{}$ to $Y_{}$ having the property that, for each $1 \leq i \leq n$, there exists either a homotopy from $f_{i-1}$ to $f_{i}$ which is constant along $K$, or a homotopy from $f_{i}$ to $f_{i-1}$ which is constant along $K$.
Suppose that $Y_{}$ is a Kan complex. Then $f$ and $g$ are homotopic relative to $K$ if and only if there exists a homotopy from $f$ to $g$ which is constant along $K$.

Proof. Set $\overline{f} = f|_{K}$. Without loss of generality, we may assume that $\overline{f}$ is also equal to $g|_{K}$. The first assertion follows by applying Remark 1.2.1.23 to the simplicial set $Z = \{ \overline{f} \} \times _{ \operatorname{Fun}(K,Y) } \operatorname{Fun}(X,Y)$. If $Y_{}$ is a Kan complex, then the restriction map $\operatorname{Fun}(X, Y) \rightarrow \operatorname{Fun}(K,Y)$ is a Kan fibration (Corollary 3.1.3.3), so that $Z$ is a Kan complex (Remark 3.1.1.9). The second assertion now follows from Proposition 1.2.5.10. $\square$