# Kerodon

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Proposition 7.6.6.13. Suppose we are given a diagram of simplicial sets

$K(0) \rightarrow K(1) \rightarrow K(2) \rightarrow K(3) \rightarrow \cdots$

having colimit $K$. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, corresponding to a compatible sequence of diagrams $f_ n: K(n) \rightarrow \operatorname{\mathcal{C}}$. Suppose that each of the diagrams $f_ n$ admits a limit in $\operatorname{\mathcal{C}}$. Then there exists a tower $X: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ with the following properties:

$(1)$

For each $n \geq 0$, the object $X(n) \in \operatorname{\mathcal{C}}$ is a limit of the diagram $f_ n$.

$(2)$

An object of $\operatorname{\mathcal{C}}$ is a limit of the diagram $f$ if and only if it is a limit of the tower $X$. In particular, the diagram $f$ has a limit if and only if the tower $X$ has a limit.

$(3)$

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories which preserves the limits of each of the diagrams $f_ n$. Then $F$ preserves limits of the diagram $f$ if and only if it preserves limits of the tower $X$.