Editing Equivalence of categories (section)

==Alternative characterizations==
A functor ''F'' : ''C'' → ''D'' yields an equivalence of categories if and only if it is simultaneously:
* [[full functor|full]], i.e. for any two objects ''c''<sub>1</sub> and ''c''<sub>2</sub> of ''C'', the map Hom<sub>''C''</sub>(''c''<sub>1</sub>,''c''<sub>2</sub>) → Hom<sub>''D''</sub>(''Fc''<sub>1</sub>,''Fc''<sub>2</sub>) induced by ''F'' is [[surjective]];
* [[faithful functor|faithful]], i.e. for any two objects ''c''<sub>1</sub> and ''c''<sub>2</sub> of ''C'', the map Hom<sub>''C''</sub>(''c''<sub>1</sub>,''c''<sub>2</sub>) → Hom<sub>''D''</sub>(''Fc''<sub>1</sub>,''Fc''<sub>2</sub>) induced by ''F'' is [[injective]]; and
* [[essentially surjective functor|essentially surjective (dense)]], i.e. each object ''d'' in ''D'' is isomorphic to an object of the form ''Fc'', for ''c'' in ''C''.<ref>Mac Lane (1998), Theorem IV.4.1</ref>
This is a quite useful and commonly applied criterion, because one does not have to explicitly construct the "inverse" ''G'' and the natural isomorphisms between ''FG'', ''GF'' and the identity functors. On the other hand, though the above properties guarantee the ''existence'' of a categorical equivalence (given a sufficiently strong version of the [[axiom of choice]] in the underlying set theory), the missing data is not completely specified, and often there are many choices. It is a good idea to specify the missing constructions explicitly whenever possible.
Due to this circumstance, a functor with these properties is sometimes called a '''weak equivalence of categories'''. (Unfortunately this conflicts with terminology from [[homotopy theory]].)

There is also a close relation to the concept of [[adjoint functors]] <math>F\dashv G</math>, where we say that <math>F:C\rightarrow D</math> is the left adjoint of <math>G:D\rightarrow C</math>, or likewise, ''G'' is the right adjoint of ''F''.  Then ''C'' and ''D'' are equivalent (as defined above in that there are natural isomorphisms from ''FG'' to '''I'''<sub>''D''</sub> and '''I'''<sub>''C''</sub> to ''GF'') if and only if <math>F\dashv G</math> and both ''F'' and ''G'' are full and faithful.

When adjoint functors <math>F\dashv G</math> are not both full and faithful, then we may view their adjointness relation as expressing a "weaker form of equivalence" of categories. Assuming that the natural transformations for the adjunctions are given, all of these formulations allow for an explicit construction of the necessary data, and no choice principles are needed. The key property that one has to prove here is that the ''counit'' of an adjunction is an isomorphism if and only if the right adjoint is a full and faithful functor.