Variant 4.6.1.3 (Endomorphism Spaces). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. We let $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ denote the simplicial set
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X) = \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{Fun}( \Delta ^1 / \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}}). \]
We will refer to $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ as the space of endomorphisms of $X$. Note that vertices of the simplicial set $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ can be identified with endomorphisms of $X$, in the sense of Definition 1.4.1.5