Tensor product of sets The cartesian product of two sets $A$ and $B$ can be seen as a tensor product.
Are there examples for the tensor product of two sets $A$ and $B$ other than the usual cartesian product ?
The context is the following: assume one has a set-valued presheaf $F$ on a monoidal category, knowing $F(A)$ and $F(B)$ how does one define $F(A \otimes B)$ ?

Simply being in a monoidal category is a rather liberal condition on the tensor product; it tells you very little about what the tensor product actually looks like.
Here is a (perhaps slightly contrived?) example:
Let $C$ be the category of vector spaces over a finite field $\mathbb F_p$ with linear transformations. The vector space tensor product makes this into a monoidal category with $\mathbb F_p$ itself as the unit. $C^{op}$ is then also a monoidal category, and the ordinary forgetful functor is a Set-valued presheaf on $C^{op}$.
However, $F(A\otimes B)$ cannot be the cartesian product $F(A)\times F(B)$, because $F(A)\times F(B)$ has the wrong cardinality when $A$ and $B$ are finite.
