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