Category Theory/Adjoint functors

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Definition (adjoint functors):

Let be categories. A pair of adjoint functors consists of two functors and (where is the left adjoint and is the right adjoint) such that the two bifunctors


from to are naturally isomorphic to each other.

Proposition (left adjoint functors preserve epimorphisms):

Let be categories, and let and be an adjoint pair of functors. Suppose that and is an epimorphism. Then is also an epimorphism.

Proof: Let be arrows in so that .

Proposition (right adjoint functors preserve monomorphisms):