< Category Theory
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):
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