Remark 5.3.3.5. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. Let $K$ be an auxiliary simplicial set, and define $\mathscr {F}^{K}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ by the formula $\mathscr {F}^{K}(C) = \operatorname{Fun}(K, \mathscr {F}(C) )$. Then the weighted nerves of $\mathscr {F}$ and $\mathscr {F}^{K}$ are related by a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }^{ \mathscr {F}^{K} }(\operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}(K, \operatorname{N}_{\bullet }^{\mathscr {F} }(\operatorname{\mathcal{C}})) \ar [d] \\ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{Fun}(K, \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) ). } \]