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Is there a technical name for a function $f$ such that $(g \circ f)(a_1, a_2,\cdots) \rightarrow g(a_1, a_2, \cdots)$? That is, is there a name for a function $f$ such that the result of composing $g$ with $f$ is $g$ invoked with $f$'s arguments?

EDIT: Accidentally reversed the order of operations and put $f \circ g$ when I meant $g \circ f$ (i.e. $g(f(\cdots)$).

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    [Identity operator](http://en.wikipedia.org/wiki/Identity_function)?2011-11-08
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    Wouldn't $f$ have to be the identity function $f(a_1,a_2,\ldots)=(a_1,a_2,\ldots)$?2011-11-08
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    $f$ is forced to be the identity only if we require $g\circ f=g$ for *all* $g$. If this identity is only required to hold for a single $g$ (which is not injective) it is possible for $f$ to differ from the identity.2011-11-08
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    Example: $g(x,y)=(1,1)$ then for any $f(x,y)$ we have $g\circ f(x,y)=g(f(x,y))=(1,1)=g(x,y)$.2011-11-08

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