# Kerodon

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Corollary 7.1.7.17. Let $K$ be a weakly contractible simplicial set and let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. If $U$ is a left fibration, then it creates $K$-indexed colimits. If $U$ is a right fibration, then it creates $K$-indexed limits.

Proof. Assume $U$ is a right fibration; we will show that it creates $K$-indexed limits (the analogous statement for left fibrations follows by a similar argument). Let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram and suppose that $U \circ f$ can be extended to a limit diagram $g: K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$. Since the inclusion $K \hookrightarrow K^{\triangleleft }$ is right anodyne (Example 4.3.7.10), our assumption that $U$ is a right fibration guarantees that the lifting problem

$\xymatrix@R =50pt@C=50pt{ K \ar [d] \ar [r]^-{f} & \operatorname{\mathcal{C}}\ar [d]^{U} \\ K^{\triangleleft } \ar [r]^-{g} \ar@ {-->}[ur]^{ \overline{f} } & \operatorname{\mathcal{D}}}$

has a solution. Since $K$ is weakly contractible, the morphism $\overline{f}$ is automatically a $U$-limit diagram (Example 7.1.7.10). Applying Corollary 7.1.7.16, we see that $\overline{f}$ is a limit diagram. $\square$