Editing Isomorphism of categories (section)

{{Use American English|date = January 2019}}
{{Short description|Relation of categories in category theory}}
In [[category theory]], two categories ''C'' and ''D'' are '''isomorphic''' if there exist [[functor]]s ''F'' : ''C'' → ''D'' and ''G'' : ''D'' → ''C'' that are mutually inverse to each other, i.e. ''FG'' = 1<sub>''D''</sub> (the identity functor on ''D'') and ''GF'' = 1<sub>''C''</sub>.<ref name="catswork">{{cite book |last=Mac Lane |first=Saunders |title=[[Categories for the Working Mathematician]] |publisher=Springer-Verlag |year=1998 |edition=2nd |series=Graduate Texts in Mathematics | volume=5 |author-link=Saunders Mac Lane |isbn=0-387-98403-8 | mr=1712872 | page=14}}</ref> This means that both the [[object (category theory)|object]]s and the [[morphism]]s of ''C'' and ''D'' stand in a [[one-to-one correspondence]] to each other. Two isomorphic categories share all properties that are defined solely in terms of category theory; for all practical purposes, they are identical and differ only in the notation of their objects and morphisms.

Isomorphism of categories is a very strong condition and rarely satisfied in practice. Much more important is the notion of [[equivalence of categories]]; roughly speaking, for an equivalence of categories we don't require that <math>FG</math> be ''equal'' to <math>1_D</math>, but only ''[[natural transformation|naturally isomorphic]]'' to <math>1_D</math>, and likewise that <math>GF</math> be naturally isomorphic to <math>1_C</math>.