Kerodon

Example 8.6.9.7 (Product Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and suppose that $\operatorname{\mathcal{D}}$ admits pairwise products. Then every pair of functors $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $\mathscr {F}': \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}$ determines a functor $\mathscr {K}: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}$, given by the formula $\mathscr {K}(C,C') = \mathscr {F}(C) \times \mathscr {F}'(C')$. In this case, we denote the coend of $\mathscr {K}$ (if it exists) by $\operatorname{Coend}( \mathscr {F}, \mathscr {F}' )$, and refer to it as the coend of $\mathscr {F}$ against $\mathscr {F}'$.