Cartesian categories with an internal Hom functor that is an adjoint functor to the product are called Cartesian closed categories.
[1] Cartesian monoidal categories have a number of special and important properties, such as the existence of diagonal maps Δx : x → x ⊗ x and augmentations ex : x → I for any object x.
In fact, any object in a cartesian monoidal category becomes a comonoid in a unique way.
Cartesian monoidal categories: Cocartesian monoidal categories: In each of these categories of modules equipped with a cocartesian monoidal structure, finite products and coproducts coincide (in the sense that the product and coproduct of finitely many objects are isomorphic).
Or more formally, if f : X1 ∐ ... ∐ Xn → X1 × ... × Xn is the "canonical" map from the n-ary coproduct of objects Xj to their product, for a natural number n, in the event that the map f is an isomorphism, we say that a biproduct for the objects Xj is an object
If, in addition, the category in question has a zero object, so that for any objects A and B there is a unique map 0A,B : A → 0 → B, it often follows that pk ∘ ij = : δij, the Kronecker delta, where we interpret 0 and 1 as the 0 maps and identity maps of the objects Xj and Xk, respectively.