Kerodon

Remark 5.5.1.6. Let $X$ and $Y$ be Kan complexes, and let $f,g: X \rightarrow Y$ be morphisms. Then $f$ and $g$ are homotopic as morphisms of simplicial sets (that is, they belong to the same connected component of the Kan complex $\operatorname{Fun}(X,Y)$) if and only if they are homotopic as morphisms in the $\infty $-category $\operatorname{\mathcal{S}}$ (Definition 1.4.3.1). Consequently, the category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Construction 3.1.5.10 can be identified with the homotopy category of the $\infty $-category $\operatorname{\mathcal{S}}$ (this is a special case of Proposition 2.4.6.9). Moreover, this identification is compatible (via the homotopy equivalences of Remark 5.5.1.5) with the $\mathrm{h} \mathit{\operatorname{Kan}}$-enrichments supplied by Remark 3.1.5.12 and Construction 4.6.9.13 (see Corollary 4.6.9.20).