Editing Equivalence of categories (section)

==Properties==
As a rule of thumb, an equivalence of categories preserves all "categorical" concepts and properties. If ''F'' : ''C'' → ''D'' is an equivalence, then the following statements are all true:
* the object ''c'' of ''C'' is an [[initial object]] (or [[terminal object]], or [[zero object]]), [[if and only if]] ''Fc'' is an [[initial object]] (or [[terminal object]], or [[zero object]]) of ''D''
* the morphism α in ''C'' is a [[monomorphism]] (or [[epimorphism]], or [[isomorphism]]), if and only if ''Fα'' is a monomorphism (or epimorphism, or isomorphism) in ''D''.
* the functor ''H'' : ''I'' → ''C'' has [[limit (category theory)|limit]] (or colimit) ''l'' if and only if the functor ''FH'' : ''I'' → ''D'' has limit (or colimit) ''Fl''. This can be applied to [[equaliser (mathematics)|equalizers]], [[product (category theory)|product]]s and [[coproduct]]s among others. Applying it to [[kernel (category theory)|kernel]]s and [[cokernel]]s, we see that the equivalence ''F'' is an [[Regular category#Exact sequences and regular functors|exact functor]].
* ''C'' is a [[cartesian closed category]] (or a [[topos]]) if and only if ''D'' is cartesian closed (or a topos).

Dualities "turn all concepts around": they turn initial objects into terminal objects, monomorphisms into epimorphisms, kernels into cokernels, limits into colimits etc.

If ''F'' : ''C'' → ''D'' is an equivalence of categories, and ''G''<sub>1</sub> and ''G''<sub>2</sub> are two inverses of ''F'', then ''G''<sub>1</sub> and ''G''<sub>2</sub> are naturally isomorphic.

If ''F'' : ''C'' → ''D'' is an equivalence of categories, and if ''C'' is a [[preadditive category]] (or [[additive category]], or [[abelian category]]), then ''D'' may be turned into a preadditive category (or additive category, or abelian category) in such a way that ''F'' becomes an [[additive functor]]. On the other hand, any equivalence between additive categories is necessarily additive. (Note that the latter statement is not true for equivalences between preadditive categories.)

An '''auto-equivalence''' of a category ''C'' is an equivalence ''F'' : ''C'' → ''C''. The auto-equivalences of ''C'' form a [[group (mathematics)|group]] under composition if we consider two auto-equivalences that are naturally isomorphic to be identical. This group captures the essential "symmetries" of ''C''. (One caveat: if ''C'' is not a small category, then the auto-equivalences of ''C'' may form a proper [[class (set theory)|class]] rather than a [[Set (mathematics)|set]].)