Kerodon

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Remark 5.2.6.8 (Functoriality in $\overrightarrow {X}$). Let $n \geq 0$ be an integer. Then the mapping simplex construction $\overrightarrow {X} \mapsto M( \overrightarrow {X} )$ determines a functor from the category $\operatorname{Fun}( [n], \operatorname{Set_{\Delta }})$ to the category of simplicial sets. This functor carries colimits of diagrams to colimits in the category of simplicial sets, and carries limits of diagrams to limits in the slice category $(\operatorname{Set_{\Delta }})_{ / \Delta ^ n }$.