Kerodon

Definition 1.4.1.1. Let $\operatorname{\mathcal{C}}= S_{\bullet }$ be an $\infty $-category. An object of $\operatorname{\mathcal{C}}$ is a vertex of the simplicial set $S_{\bullet }$ (that is, an element of the set $S_{0}$). A morphism of $\operatorname{\mathcal{C}}$ is an edge of the simplicial set $S_{\bullet }$ (that is, an element of $S_1$). If $f \in S_{1}$ is a morphism of $\operatorname{\mathcal{C}}$, we will refer to the object $X = d^{1}_1(f)$ as the source of $f$ and to the object $Y = d^{1}_0(f)$ as the target of $f$. In this case, we will say that $f$ is a morphism from $X$ to $Y$. For any object $X$ of $\operatorname{\mathcal{C}}$, we can regard the degenerate edge $s^{0}_0(X)$ as a morphism from $X$ to itself; we will denote this morphism by $\operatorname{id}_{X}$ and refer to it as the identity morphism of $X$.