Editing Adjoint functors (section)

==History==

The idea of adjoint functors was introduced by [[Daniel Kan]] in 1958.<ref>{{Cite journal|last=Kan|first=Daniel M.|date=1958|title=Adjoint Functors|url=https://www.ams.org/journals/tran/1958-087-02/S0002-9947-1958-0131451-0/S0002-9947-1958-0131451-0.pdf|journal=[[Transactions of the American Mathematical Society]]|volume=87|issue=2|pages=294–329|doi=10.2307/1993102|jstor=1993102 |doi-access=free}}</ref>  Like many of the concepts in category theory, it was suggested by the needs of [[homological algebra]], which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as

:hom(''F''(''X''), ''Y'') = hom(''X'', ''G''(''Y''))

in the category of [[abelian group]]s, where ''F'' was the functor <math>- \otimes A</math> (i.e. take the [[tensor product]] with ''A''), and ''G'' was the functor hom(''A'',–) (this is now known as the [[tensor-hom adjunction]]).
<!--Here hom(''X'',''Y'') means 'all [[group homomorphism|homomorphisms]] of [[abelian group]]s'.-->
The use of the ''equals'' sign is an [[abuse of notation]]; those two groups are not really identical but there is a way of identifying them that is ''natural''. It can be seen to be natural on the basis, firstly, that these are two alternative descriptions of the [[bilinear mapping]]s from ''X'' × ''A'' to ''Y''. That is, however, something particular to the case of tensor product. In category theory the 'naturality' of the bijection is subsumed in the concept of a [[natural isomorphism]].