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Notation 7.6.6.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will generally abuse notation by identifying a functor $X: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{C}}$ with the collection of objects $\{ X(n) \} _{n \geq 0}$ and morphisms $f_{n}: X(n+1) \rightarrow X(n)$ obtained by the evaluating $X$ on the edges of $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ corresponding to ordered pairs of the form $(n,n+1)$; we will depict the pair $( \{ X(n) \} _{n \geq 0}, \{ f_ n \} _{n \geq 0} )$ as a diagram

\[ X(0) \xrightarrow { f_0 } X(1) \xrightarrow { f_1} X(2) \xrightarrow { f_2} X(3) \xrightarrow { f_3} X(4) \rightarrow \cdots \]

Similarly, we abuse notation by identifying towers $X: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ with diagrams

\[ \cdots \rightarrow X(4) \xrightarrow { f_3 } X(3) \xrightarrow { f_2 } X(2) \xrightarrow { f_1} X(1) \xrightarrow { f_0 } X(0). \]