Adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. By definition, an adjunction between categories C and D is a pair of functors
$$ F: \mathcal{D} \rightarrow \mathcal{C} \text { and } G: \mathcal{C} \rightarrow \mathcal{D} $$and, for all objects $X$ in $C$ and $Y$ in $D$ a bijection between the respective morphism sets
$$ \operatorname{hom}_{\mathcal{C}}(F Y, X) \cong \operatorname{hom}_{\mathcal{D}}(Y, G X) $$such that this family of bijections is natural in $X$ and $Y$. For more info (the meaning of natural, here, for example), see the wikipedia page Adjoint functors.
The left part of a pair of adjoint functors is one of two best approximations to a weak inverse of the other functor of the pair. (The other best approximation is the functor's right adjoint, if it exists. ) Note that a weak inverse itself, if it exists, must be a left adjoint, forming an adjoint equivalence.
A left adjoint to a forgetful functor is called a free functor; in general, left adjoints may be thought of as being defined freely, consisting of anything that an inverse might want, regardless of whether it works.
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Author of the notes: Antonio J. Pan-Collantes
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