Definition:Adjunction

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Definition

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 $(F,G,\alpha)$, where

$F : \mathbf D \to \mathbf C$ is a functor.
$G : \mathbf C \to \mathbf D$ is a functor.
$\alpha : \mathrm{Hom}_{\mathbf C}(F-,-) \to \mathrm{Hom}_{\mathbf D}(-,G-)$ is a natural isomorphism between the functors
$\mathrm{Hom}_{\mathbf C}(F-,-) : \mathbf D^{\mathrm{op}} \times \mathbf D \to \mathbf{Set}$
$\mathrm{Hom}_{\mathbf D}(-,G-) : \mathbf D^{\mathrm{op}} \times \mathbf D \to \mathbf{Set}$