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

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

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

Proposition (left adjoint functors preserve epimorphisms):

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

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

Proposition (right adjoint functors preserve monomorphisms):