Notation 8.5.6.11. Let $q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \Delta ^1 / \operatorname{\partial \Delta }^1$ be the covering map of Remark 8.5.4.13. For every $\infty $-category $\operatorname{\mathcal{C}}$, precomposition with $q$ induces a functor
\[ T: \operatorname{End}_{\operatorname{\mathcal{C}}} = \operatorname{Fun}( \Delta ^1 / \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{C}}) \quad \quad (X,e) \mapsto T_{e}. \]
More informally, the functor $T$ carries each endomorphism $e: X \rightarrow X$ in the $\infty $-category $\operatorname{\mathcal{C}}$ to the associated sequential diagram
\[ \cdots \rightarrow X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \rightarrow \cdots \]