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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:
$\map {\mathrm {Hom}_{\mathbf C} } {F-, -} : \mathbf D^{\mathrm {op}} \times \mathbf D \to \mathbf{Set}$
$\map {\mathrm {Hom}_{\mathbf D} } {-, G-} : \mathbf D^{\mathrm {op} } \times \mathbf D \to \mathbf{Set}$