If function composition is the sum of functions' universe,
$$f\oplus g :=fog,$$
then we can also define difference in the universe of functions
$$f\ominus g:=fog^{-1}$$
To define the analogue of product we need to repeat sum and then make it a binary relation.
$$f^n :=\underset{\text{n times}}{fofo\cdots of}$$
We next need to turn \(n\) to a function.
$$(fpg)(x) :=\underbrace{fof\cdots of}_{g(x) \text{times}}$$
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