Kerodon

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Remark 2.4.1.9. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a locally Kan simplicial category, and let $f,g: X \rightarrow Y$ be a pair of morphisms in the underlying category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$ having the same source and target. Invoking Proposition 1.1.9.10, we see that the following conditions are equivalent:

$(a)$

There exists a homotopy from $f$ to $g$, in the sense of Definition 2.4.1.6.

$(b)$

The morphisms $f$ and $g$ belong to the same connected component of the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$.

In particular, condition $(a)$ defines an equivalence relation on the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.