# Kerodon

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Example 7.5.3.13 (Towers of Isofibrations). Suppose we are given a tower of $\infty$-categories

$\cdots \rightarrow \operatorname{\mathcal{C}}(3) \rightarrow \operatorname{\mathcal{C}}(2) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \operatorname{\mathcal{C}}(0),$

which we identify with a functor $\mathscr {F}: \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} \rightarrow \operatorname{QCat}$. If each of the transition functors $\operatorname{\mathcal{C}}(n+1) \rightarrow \operatorname{\mathcal{C}}(n)$ is an isofibration, then the comparison map $\varprojlim _{n} \operatorname{\mathcal{C}}(n) = \varprojlim (\mathscr {F}) \hookrightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ is an equivalence of $\infty$-categories. This follows by combining Example 4.5.6.8 with Proposition 7.5.3.12.