Can we find a non-cartesian monoidal category where for each pair of objects the tensor product $A \otimes B$ is a cartesian product of $A$ and $B$?

Let me explain.

A [monoidal category](https://en.wikipedia.org/wiki/Monoidal_category#Formal_definition) $(\mathsf{C}, \otimes, I, a, r, \ell)$ is a category $\mathsf{C}$ with a tensor product $\otimes$, a unit object $I$, an associator $a$, and left and right unitors $r$ and $\ell$ obeying [the pentagon and triangle identities](https://math.ucr.edu/home/baez/qg-winter2001/definitions.pdf).  A cartesian category is one with [binary products](https://en.wikipedia.org/wiki/Product_(category_theory)#Definition) and a [terminal object](https://en.wikipedia.org/wiki/Initial_and_terminal_objects).

Given a cartesian category $\mathsf{C}$, and choosing a terminal object, and choosing a product cone 

$$ A \leftarrow A \times B \rightarrow B$$

for each pair of objects $A,B$, there is a well-known way to make $\mathsf{C}$ into a monoidal category. To create this monoidal category, first we take $A \otimes B$ to be the object $A \times B$ in the chosen product cone for $A$ and $B$.   Then we define $\otimes$ on morphisms using the universal property of the chosen product cones.  Similarly, we define the associator

$$ a_{A,B,C} \colon (A \otimes B) \otimes C \to A \otimes (B \otimes C)$$ 

using the universal property of the chosen product cones.  Then we take the unit object $I$ to be the chosen terminal object, and define the left and right unitors

$$ \ell_A \colon I \otimes A \to A, \qquad r_A \colon A \otimes I \to A$$

using the universal property of the chosen product cones and the fact that $I$ is terminal.   We obtain a monoidal category $(\mathsf{C}, \otimes, I, a, r, \ell)$ called a 'cartesian monoidal category'.

More generally, any monoidal category *equivalent as a monoidal category* to one of this form is called a **[cartesian monoidal category](https://ncatlab.org/nlab/show/cartesian+monoidal+category)**.     

This leaves open the possibility that there could be a way to make our cartesian category $\mathsf{C}$ into a monoidal category where for each pair $A, B \in \mathsf{C}$ the object $A \otimes B$ equal to the chosen object $A \times B$, but which however is *not equivalent*, as a monoidal category, to the cartesian monoidal category described above.   **Is this ever possible?**

There are various things we can change: the action of $\otimes$ on morphisms, the associator, the left and right unitors - and *prima facie*, given the way I've worded the question, even the unit object $I$.

It would be especially fun if we could do this starting from some familiar cartesian category like the category of sets.

[Edit: I have a conjecture [here](https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/Monoidal.20category.20question/near/453810648) for how we might get a noncartesian monoidal category where $A \otimes B$ is the product of $A$ and $B$ for every pair of objects $A,B$.]