Variant 7.6.4.9. Let $\operatorname{\mathbf{Z}}_{\geq 0}$ denote the collection of nonnegative integers, which we regard as a commutative monoid under addition, and let $B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ denote the classifying simplicial set of Construction 1.3.2.5. The simplicial set $B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ is an $\infty $-category which contains a (unique) object $X$, and the generator $1 \in \operatorname{\mathbf{Z}}_{\geq 0}$ determines an endomorphism $e: X \rightarrow X$. We can regard $B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ as freely generated by the endomorphism $e$: more precisely, the pair $(X,e)$ determines a morphism of simplicial sets $\sigma : \Delta ^1 / \operatorname{\partial \Delta }^1 \hookrightarrow B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ which is inner anodyne (see Example 1.5.7.11), and therefore induces a trivial Kan fibration $\operatorname{Fun}( B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1 / \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})$ for every $\infty $-category $\operatorname{\mathcal{C}}$. In particular, the morphism $\sigma $ is both left and right cofinal (Proposition 7.2.1.3).
If $F: B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0} \rightarrow \operatorname{\mathcal{C}}$ is a functor of $\infty $-categories, then Corollary 7.2.2.11 guarantees that an object of $\operatorname{\mathcal{C}}$ is a limit of the functor $F$ if and only if it is a limit of the diagram $F \circ \sigma $: that is, if and only if it is an equalizer of the pair of morphisms $F(e), \operatorname{id}_{F(X)}: F(X) \rightarrow F(X)$ (see Example 7.6.4.8). Similarly, an object of $\operatorname{\mathcal{C}}$ is a colimit of the functor $F$ if and only if it is a coequalizer of the pair $( F(e), \operatorname{id}_{ F(X)} )$.