Any cocartesian closed category is equivalent to the terminal category.
Let 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 and the set is in bijection with where is the unit object and the internal-hom. But since is an initial object, is a singleton, hence so is .
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).
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 will be itself initial (as left adjoints preserve colimits). Furthermore, given any morphism from an object to , we can pair this with the identity function on to obtain a morphism from into with a left inverse given by projection, thus identifying as a retract of an initial object, and therefore as an initial object itself. It follows immediately that any two parallel maps to are equal; equivalently, by the dual reasoning, any two parallel maps out of are equal. But this means any two parallel maps in general are equal, as maps from to can be identified with maps from to ; accordingly, the category must be a preorder.