nLab
cocartesian closed category

A cocartesian closed category is a cocartesian monoidal category which is a closed monoidal category. The notion is not very interesting, however, because of the following:

Theorem

Any cocartesian closed category is equivalent to the terminal category.

Proof

Let $C$ be cocartesian closed. Since it has an initial object, it is inhabited; thus it suffices to show that between any two objects there is a unique morphism. But in any closed monoidal category, for any objects $x$ and $y$ the set $C(x,y)$ is in bijection with $C(I,[x,y])$ where $I$ is the unit object and $[-,-]$ the internal-hom. But since $I$ is an initial object, $C(I,[x,y])$ is a singleton, hence so is $C(x,y)$.

Note that the proof actually applies to any semi-cocartesian closed monoidal category, i.e. any closed monoidal category whose unit is an initial object.

On the other hand, there are many co-cartesian co-closed? categories, namely the opposite category of any cartesian closed category. Any category which is both cartesian closed and co-cartesian co-closed is a preorder, though it may not be the terminal category (e.g., any Boolean algebra is such a category).

Proof

Let a category be given which is both cartesian closed and co-(cartesian closed). Cartesian closure tells us that the product of any object with the initial object $0$ will be itself initial (as left adjoints preserve colimits). Furthermore, given any morphism from an object $A$ to $0$, we can pair this with the identity function on $A$ to obtain a morphism from $A$ into $A\times 0$ with a left inverse given by projection, thus identifying $A$ as a retract of an initial object, and therefore as an initial object itself. It follows immediately that any two parallel maps to $0$ are equal; equivalently, by the dual reasoning, any two parallel maps out of $1$ are equal. But this means any two parallel maps in general are equal, as maps from $X$ to $Y$ can be identified with maps from $1$ to $Y^X$; accordingly, the category must be a preorder.