Distributive category

In mathematics, a category is distributive if it has finite products and finite coproducts such that for every choice of objects formula_1, the canonical map
is an isomorphism, and for all objects formula_3, the canonical map formula_4 is an isomorphism (where 0 denotes the initial object). Equivalently, if for every object formula_3 the endofunctor formula_6 defined by formula_7 preserves coproducts up to isomorphisms formula_8. It follows that formula_8 and aforementioned canonical maps are equal for each choice of objects. 
In particular, if the functor formula_6 has a right adjoint (i.e., if the category is cartesian closed), it necessarily preserves all colimits, and thus any cartesian closed category with finite coproducts (i.e., any bicartesian closed category) is distributive.
Example.
The category of sets is distributive. Let , , and be sets. Then
where formula_12 denotes the coproduct in Set, namely the disjoint union, and formula_13 denotes a bijection. In the case where , , and are finite sets, this result reflects the distributive property: the above sets each have cardinality formula_14.
The categories Grp and Ab are not distributive, even though they have both products and coproducts.
An even simpler category that has both products and coproducts but is not distributive is the category of pointed sets.