Kerodon

Proof. The category $\operatorname{Cat}$ admits small limits and colimits, which are preserved by the forgetful functor $\operatorname{Ob}: \operatorname{Cat}\rightarrow \operatorname{Set}$. It follows that the category $\operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Cat})$ of simplicial objects in $\operatorname{Cat}$ also admits small limits and colimits, which are computed pointwise. Remark 2.4.1.12 supplies a fully faithful embedding $\operatorname{Cat_{\Delta }}\hookrightarrow \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Cat})$ whose essential image is closed under small limits and colimits, so that $\operatorname{Cat_{\Delta }}$ admits small limits and colimits as well. $\square$