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Example 4.5.6.16. Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \cdots \ar [r] & \operatorname{\mathcal{C}}(3) \ar [d] \ar [r] & \operatorname{\mathcal{C}}(2) \ar [r] \ar [d] & \operatorname{\mathcal{C}}(1) \ar [r] \ar [d] & \operatorname{\mathcal{C}}(0) \ar [d] \\ \cdots \ar [r] & \operatorname{\mathcal{D}}(3) \ar [r] & \operatorname{\mathcal{D}}(2) \ar [r] & \operatorname{\mathcal{D}}(1) \ar [r] & \operatorname{\mathcal{D}}(0), } \]

where the horizontal maps are isofibrations and the vertical maps are equivalences of $\infty $-categories. Then the induced map $\varprojlim \operatorname{\mathcal{C}}(n) \rightarrow \varprojlim \operatorname{\mathcal{D}}(n)$ is an equivalence of $\infty $-categories. This follows by combining Example 4.5.6.7, Corollary 4.5.6.11, and Corollary 4.5.6.15.