\begin{definition}[Definition:Adjunction]
Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$, $\mathbf D$ be locally small categories.
An '''adjunction''' between $\mathbf C$ and $\mathbf D$ is a triple $\tuple {F, G, \alpha}$, where
:$F : \mathbf D \to \mathbf C$ is a functor.
:$G : \mathbf C \to \mathbf D$ is a functor.
:$\alpha: \map {\mathrm {Hom}_{\mathbf C} } {F-, -} \to \map {\mathrm {Hom}_{\mathbf D} } {-, G-}$ is a natural isomorphism between the functors:
::$\map {\mathrm {Hom}_{\mathbf C} } {F-, -} : \mathbf D^{\mathrm {op}} \times \mathbf D \to \mathbf{Set}$
::$\map {\mathrm {Hom}_{\mathbf D} } {-, G-} : \mathbf D^{\mathrm {op} } \times \mathbf D \to \mathbf{Set}$
Category:Definitions/Category Theory
\end{definition}