Category Theory/Adjoint functors

Definition (adjoint functors):

Let ${\mathcal {A}},{\mathcal {B}}$ be categories. A pair of adjoint functors consists of two functors $L:{\mathcal {A}}\to {\mathcal {B}}$ and $R:{\mathcal {B}}\to {\mathcal {A}}$ (where $L$ is the left adjoint and $R$ is the right adjoint) such that the two bifunctors

$\operatorname {Hom} _{\mathcal {B}}(L\cdot ,\cdot )$ and $\operatorname {Hom} _{\mathcal {A}}(\cdot ,R\cdot )$

from $({\mathcal {A}},{\mathcal {B}})$ to $\operatorname {Set}$ are naturally isomorphic to each other.

Proposition (left adjoint functors preserve epimorphisms):

Let ${\mathcal {A}},{\mathcal {B}}$ be categories, and let $L:{\mathcal {A}}\to {\mathcal {B}}$ and $R:{\mathcal {B}}\to {\mathcal {A}}$ be an adjoint pair of functors. Suppose that $x,y\in A$ and $f\in \operatorname {Hom} _{\mathcal {A}}(X,Y)$ is an epimorphism. Then $Lf:LX\to LY$ is also an epimorphism.

Proof: Let $g,h:LY\to Z$ be arrows in ${\mathcal {B}}$ so that $g\circ Lf=h\circ Lf$. $\Box$

Proposition (right adjoint functors preserve monomorphisms):