# Kerodon

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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

$\cdots \rightarrow X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \rightarrow \cdots$