Kerodon

Example 7.6.4.8 (Fixed Points of Endomorphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. An endomorphism of $X$ is a morphism $f: X \rightarrow X$ from the object $X$ to itself. Note that the pair $(X,f)$ can be identified with a morphism of simplicial sets $\sigma : (\Delta ^1 / \operatorname{\partial \Delta }^1) \rightarrow \operatorname{\mathcal{C}}$. It follows from Remark 7.6.4.3 (together with Corollary 7.2.2.11) that an object of $\operatorname{\mathcal{C}}$ is a limit of the diagram $\sigma $ if and only if it is an equalizer of the pair of morphisms $f, \operatorname{id}_{X}: X \rightarrow X$. Similarly, an object of $\operatorname{\mathcal{C}}$ is a colimit of $\sigma $ if and only if it is a coequalizer of the pair $(f, \operatorname{id}_ X)$.