Editing Identity function (section)

==Algebraic properties==
If {{math|''f'' : ''X'' → ''Y''}} is any function, then {{math|1=''f'' ∘ id<sub>''X''</sub> = ''f'' = id<sub>''Y''</sub> ∘ ''f''}}, where "∘" denotes [[function composition]].<ref>{{cite book
 | last = Nel | first = Louis
 | year = 2016
 | title = Continuity Theory
 | url = https://books.google.com/books?id=_JdPDAAAQBAJ&pg=PA21
 | page = 21
 | publisher = Springer
 | location = Cham
 | doi = 10.1007/978-3-319-31159-3
 | isbn = 978-3-319-31159-3
}}</ref> In particular, {{math|id<sub>''X''</sub>}} is the [[identity element]] of the [[monoid]] of all functions from {{math|''X''}} to {{math|''X''}} (under function composition).

Since the identity element of a monoid is [[unique (mathematics)|unique]],<ref>{{Cite book|last1=Rosales|first1=J. C.|url=https://books.google.com/books?id=LQsH6m-x8ysC&q=identity+element+of+a+monoid+is+unique&pg=PA1|title=Finitely Generated Commutative Monoids|last2=García-Sánchez|first2=P. A.|date=1999|publisher=Nova Publishers|isbn=978-1-56072-670-8|pages=1|language=en|quote=The element 0 is usually referred to as the identity element and if it exists, it is unique}}</ref> one can alternately define the identity function on {{math|''M''}} to be this identity element. Such a definition generalizes to the concept of an [[identity morphism]] in [[category theory]], where the [[endomorphism]]s of {{math|''M''}} need not be functions.