Kerodon

Remark 7.5.2.3. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a (strictly commutative) diagram of $\infty $-categories, let $K$ be a simplicial set, and let $\mathscr {F}^{K}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ denote the functor given on objects by the formula $\mathscr {F}^{K}(C) = \operatorname{Fun}(K, \mathscr {F}(C) )$. Then there is a canonical isomorphism of simplicial sets $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F}^{K} ) \simeq \operatorname{Fun}(K, \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} ) )$ (see Remarks 5.3.3.5 and 5.3.1.19).