Editing Endomorphism (section)

{{Short description|Self-self morphism}}
{{redirect|Endomorphic|the Sheldon body type|Somatotype and constitutional psychology}}
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[[File:Orthogonal projection.svg|frame|right|[[Orthogonal projection]] onto a line, {{math|''m''}}, is a [[linear operator]] on the plane. This is an example of an endomorphism that is not an [[automorphism]].]]
In [[mathematics]], an '''endomorphism''' is a [[morphism]] from a [[mathematical object]] to itself.  An endomorphism that is also an [[isomorphism]] is an [[automorphism]]. For example, an endomorphism of a [[vector space]] {{math|''V''}} is a [[linear map]] {{math|''f'': ''V'' → ''V''}}, and an endomorphism of a [[group (mathematics)|group]] {{math|''G''}} is a [[group homomorphism]] {{math|''f'': ''G'' → ''G''}}. In general, we can talk about endomorphisms in any [[Category (mathematics)|category]]. In the [[category of sets]], endomorphisms are [[Function (mathematics)|functions]] from a [[Set (mathematics)|set]] ''S'' to itself.

In any category, the [[function composition|composition]] of any two endomorphisms of {{math|''X''}} is again an endomorphism of {{math|''X''}}.  It follows that the set of all endomorphisms of {{math|''X''}} forms a [[monoid]], the [[full transformation monoid]], and denoted {{math|End(''X'')}} (or {{math|End{{sub|''C''}}(''X'')}} to emphasize the category {{math|''C''}}).