Identity function
Algebraic properties

If f : X → Y is any function, then f ∘ id = f = id ∘ f, where "∘" denotes function composition. In particular, id is the identity element of the monoid of all functions from X to X (under function composition).

Since the identity element of a monoid is unique, one can alternately define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of M need not be functions.
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