# Kerodon

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Corollary 4.5.6.21. Suppose we are given an inverse system of $\infty$-categories

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

where each of the transition functors $\operatorname{\mathcal{C}}(n) \rightarrow \operatorname{\mathcal{C}}(n-1)$ is an isofibration. Then the limit $\operatorname{\mathcal{C}}= \varprojlim _{n} \operatorname{\mathcal{C}}(n)$ is an $\infty$-category, whose core $\operatorname{\mathcal{C}}^{\simeq }$ is the inverse limit $\varprojlim _{n} \operatorname{\mathcal{C}}(n)^{\simeq }$. In other words, a morphism of $\operatorname{\mathcal{C}}$ is an isomorphism if and only if its image in each $\operatorname{\mathcal{C}}(n)$ is an isomorphism.