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In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects A , B , C {\displaystyle A,B,C} , the canonical map
is an isomorphism, and for all objects A {\displaystyle A} , the canonical map 0 → A × 0 {\displaystyle 0\to A\times 0} is an isomorphism. Equivalently, if for every object A {\displaystyle A} the endofunctor A × − {\displaystyle A\times -} defined by B ↦ A × B {\displaystyle B\mapsto A\times B} preserves coproducts up to isomorphisms f {\displaystyle f}. It follows that f {\displaystyle f} and aforementioned canonical maps are equal for each choice of objects.
In particular, if the functor A × − {\displaystyle A\times -} has a right adjoint , it necessarily preserves all colimits, and thus any cartesian closed category with finite coproducts is distributive.