Let $\mathcal{C}$ be a category, let $\mathcal{F}\colon \mathcal{C}^{\mathsf{op}}\to \mathsf{Sets}$ be a presheaf on $\mathcal{C}$, and let $G\colon \mathcal{C}\to \mathsf{Sets}$ be a copresheaf on $\mathcal{C}$.
The tensor product of $\mathcal{F}$ with $G$ is the set $\mathcal{F}\boxtimes _{\mathcal{C}}G$1 defined by
In other words, the tensor product of $\mathcal{F}$ with $G$ is the set $\mathcal{F}\boxtimes _{\mathcal{C}}G$ defined as the coend of the functor
which is equivalently the composition
Let $\mathcal{C}$ be a category.
Functoriality. The assignments $\mathcal{F},G,(\mathcal{F},G)\mapsto \mathcal{F}\boxtimes _{\mathcal{C}}G$ define functors
As a Composition of Profunctors. Let $\mathcal{C}$ be a category and let:
$\mathcal{F}\colon \mathsf{pt}\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}\mathcal{C}$ be a presheaf on $\mathcal{C}$, viewed as a profunctor.
$F\colon \mathcal{C}\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}\mathsf{pt}$ be a copresheaf on $\mathcal{C}$, viewed as a profunctor.
We have a natural isomorphism of profunctors
Interaction With Representable Presheaves. Let $\mathcal{F}$ be a presheaf on $\mathcal{C}$. We have a bijection of sets
natural in $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, giving a natural isomorphism of functors
Interaction With Corepresentable Copresheaves. Let $G$ be a copresheaf on $\mathcal{C}$. We have a bijection of sets
natural in $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, giving a natural isomorphism of functors
Interaction With Yoneda Extensions. Let $G\colon \mathcal{C}\to \mathsf{Sets}$ be a copresheaf on $\mathcal{C}$. We have a natural isomorphism
Interaction With Contravariant Yoneda Extensions. Let $\mathcal{F}\colon \mathcal{C}^{\mathsf{op}}\to \mathsf{Sets}$ be a presheaf on $\mathcal{C}$. We have a natural isomorphism
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