Definition 1.3.1.1. 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 $X = d_1(f)$ as the *source* of $f$ and to the object $Y = d_0(f)$ 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 $s_0(X)$ 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 $X$.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$