category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A cartesian monoidal category (usually just called a cartesian category), is a monoidal category whose monoidal structure is given by the category-theoretic product (and so whose unit is a terminal object).
A cartesian monoidal category which is also closed is called a cartesian closed category.
A strong monoidal functor between cartesian categories is called a cartesian functor.
Any category with finite products can be considered as a cartesian monoidal category (as long as we have either (1) a specified product for each pair of objects, (2) a global axiom of choice, or (3) we allow the monoidal product to be an anafunctor).
The term cartesian category usually means a category with finite products but can also mean a finitely complete category, so we avoid that term.
Cartesian monoidal categories have a number of special and important properties, such as the existence of diagonal maps $\Delta_x : x \to x\otimes x$ and augmentations $e_x: x \to I$ for any object $x$. In applications to computer science we can think of $\Delta$ as ‘duplicating data’ and $e$ as ‘deleting’ data. These maps make any object into a comonoid. In fact, any object in a cartesian monoidal category becomes a comonoid in a unique way.
Moreover, one can show that any symmetric monoidal category equipped with suitably well-behaved diagonals and augmentations must in fact be cartesian monoidal. More precisely: suppose $C$ is a symmetric monoidal category equipped with monoidal natural transformations
and
such that the following composites are identity morphisms:
where $r$, $\ell$ are the right and left unitors. Then $C$ is cartesian.
Heuristically: a symmetric monoidal category is cartesian if we can duplicate and delete data, and ‘duplicating a piece of data and then deleting one copy is the same as not doing anything’.
A related theorem describes cartesian monoidal categories as monoidal categories satisfying two properties involving the unit object. First, we say a monoidal category $C$ is semicartesian if the unit for the tensor product is terminal. If this is true, any tensor product of objects $x \otimes y$ comes equipped with morphisms
given by
and
respectively, where $e$ stands for the unique morphism to the terminal object and $r$, $\ell$ are the right and left unitors. We can thus ask whether $p_x$ and $p_y$ make $x \otimes y$ into the product of $x$ and $y$. If so, it is a theorem that $C$ is a cartesian monoidal category. (This theorem is probably in Eilenberg and Kelly’s paper on closed categories, but they may not have been the first to note it.)