Editing Equivalence of categories (section)

==Definition==
Formally, given two categories ''C'' and ''D'', an ''equivalence of categories'' consists of a functor ''F'' : ''C'' → ''D'', a functor ''G'' : ''D'' → ''C'', and two natural isomorphisms ε: ''FG''→'''I'''<sub>''D''</sub> and η : '''I'''<sub>''C''</sub>→''GF''. Here ''FG'': ''D''→''D'' and ''GF'': ''C''→''C'' denote the respective compositions of ''F'' and ''G'', and '''I'''<sub>''C''</sub>: ''C''→''C'' and '''I'''<sub>''D''</sub>: ''D''→''D'' denote the ''identity functors'' on ''C'' and ''D'', assigning each object and morphism to itself. If ''F'' and ''G'' are contravariant functors one speaks of a ''duality of categories'' instead.

One often does not specify all the above data. For instance, we say that the categories ''C'' and ''D'' are ''equivalent'' (respectively ''dually equivalent'') if there exists an equivalence (respectively duality) between them. Furthermore, we say that ''F'' "is" an equivalence of categories if an inverse functor ''G'' and natural isomorphisms as above exist. Note however that knowledge of ''F'' is usually not enough to reconstruct ''G'' and the natural isomorphisms: there may be many choices (see example below).