# 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

- and

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)**: