Editing Adjoint functors (section)

===Equivalences of categories===

If a functor ''F'' : ''D'' → ''C'' is one half of an [[equivalence of categories]] then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms.

Every adjunction 〈''F'', ''G'', ε, η〉 extends an equivalence of certain subcategories. Define ''C''<sub>1</sub> as the full subcategory of ''C'' consisting of those objects ''X'' of ''C'' for which ε<sub>''X''</sub> is an isomorphism, and define ''D''<sub>1</sub> as the [[full subcategory]] of ''D'' consisting of those objects ''Y'' of ''D'' for which η<sub>''Y''</sub> is an isomorphism. Then ''F'' and ''G'' can be restricted to ''D''<sub>1</sub> and ''C''<sub>1</sub> and yield inverse equivalences of these subcategories.

In a sense, then, adjoints are "generalized" inverses. Note however that a right inverse of ''F'' (i.e. a functor ''G'' such that ''FG'' is naturally isomorphic to 1<sub>''D''</sub>) need not be a right (or left) adjoint of ''F''. Adjoints generalize ''two-sided'' inverses.