# Kerodon

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Remark 1.1.4.6. Let $\operatorname{\mathcal{C}}$ be a category. Proposition 1.1.4.5 can be rephrased as follows:

• For any simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$, the limit $\varprojlim _{ [n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} } X_{n}$ exists in the category $\operatorname{\mathcal{C}}$.

• The canonical map $\varprojlim _{ [n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} } X_{n} \rightarrow X_0$ is an isomorphism.

These assertions follow formally from the observation that $[0]$ is a final object of the category $\operatorname{{\bf \Delta }}$ (and therefore an initial object of the category $\operatorname{{\bf \Delta }}^{\operatorname{op}}$).