The tensor product of a presheaf and a covariant set-valued functor

Let $\mathcal{C}$ be a category, let $P:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$ be a presheaf (Presheaves, covariantly and contravariantly representable functors, and representations, Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$) and let $F:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ be a covariant functor (Sets and functions form the large locally small category $\mathbf{S}\mathbf{e}\mathbf{t}$). The assignment

$T(c_1,c_2):=P(c_1)\times F(c_2)$

is a functor ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ (Product category and its projection functors, The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$): it is contravariant in $c_1$ because $P$ is, covariant in $c_2$ because $F$ is, and the two slots act independently, on the two coordinates of the Cartesian product.

The tensor product of $P$ and $F$ over $\mathcal{C}$ is the coend of the product of a presheaf and a covariant set-valued functor (The end and the coend of a functor ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$), when it exists:

$P{\otimes }_{\mathcal{C}}F:={\int }^{c}P(c)\times F(c)\mathrm{.}$

Its cowedge components are written ${\rho }_c:P(c)\times F(c)\to P{\otimes }_{\mathcal{C}}F$, and the cowedge equation reads ${\rho }_c(P(f)(y),x)={\rho }_{c^{\mathrm{\prime }}}(y,F(f)(x))$ for $f:c\to c^{\mathrm{\prime }}$, $y\in P(c^{\mathrm{\prime }})$ and $x\in F(c)$.

The variance is written into the definition rather than left to the reader. A coend needs its integrand contravariant in the first slot and covariant in the second, so a presheaf and a covariant functor are exactly the pair for which the displayed product is an integrand; two covariant functors do not give one, and the expression ${\int }^{c}F(c)\times G(c)$ for two covariant $F$ and $G$ is not defined.

The name records the analogy with a tensor product of modules: the cowedge equation moves an element of $\mathcal{C}$ across the product exactly as a scalar moves across ${\otimes }_R$, and The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category supplies the actions when $P$ and $F$ are hom-functors. The analogy is made precise for a one-object $\mathcal{C}$ on this page's companion, where the two functors are a right and a left action of a monoid.