Kerodon

Remark 4.6.5.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of morphisms $f,g: X \rightarrow Y$ having the same source and target. Then the datum of an edge $e: f \rightarrow g$ in the left-pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ is equivalent to the datum of a homotopy from $f$ to $g$, in the sense of Definition 1.4.3.1. In particular, $f$ and $g$ are homotopic if and only if they belong to the same connected component of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$. We therefore have a canonical bijection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \simeq \pi _0( \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y) )$.