Let $\mathcal{C}$ be a category, let $\mathcal{D}$ be a category with coproducts, let $\mathcal{F}\colon \mathcal{C}^{\mathsf{op}}\to \mathsf{Sets}$ be a presheaf on $\mathcal{C}$, and let $G\colon \mathcal{C}\to \mathcal{D}$ be a functor.
The tensor of $\mathcal{F}$ with $G$ is the object $\mathcal{F}\odot _{\mathcal{C}}G$1 of $\mathcal{D}$ defined by
In other words, the tensor of $\mathcal{F}$ with $G$ is the object $\mathcal{F}\odot _{\mathcal{C}}G$ of $\mathcal{D}$ defined as the coend of the functor
Let $\mathcal{C}$ be a category.
Functoriality. The assignments $\mathcal{F},G,(\mathcal{F},G)\mapsto \mathcal{F}\odot _{\mathcal{C}}G$ define functors
Interaction With Corepresentable Copresheaves. We have an isomorphism
natural in $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, giving a natural isomorphism of functors
Interaction With Yoneda Extensions. We have a natural isomorphism
natural in $G\in \operatorname {\mathrm{Obj}}(\mathsf{Fun}(\mathcal{C},\mathcal{D}))$.
: Interaction With Corepresentable Copresheaves.
