Functional super-iteration and hierarchy of functional hyper-iterations

[Base-Acid Tetration]

bo, you needed to see my explanation.

1. Evaluate g(x) at c first.
2. Hyper-n-iterate f to the OUTPUT of Step 1.
3. Evaluate the resulting function at c.

For n=1 (iteration), to evaluate this expression at any given natural x=c:
1. Evaluate g(x) at c first.
2. Iterate f to the OUTPUT of Step 1.
3. Evaluate the resulting function at c.

The set of all ordered pairs resulting from this evaluation is {(x, [f It_1 g(x)] (x))}.

I know, but I dont know what you want to say with that *questioningly look*.
We agree on the meaning, the question is more how to write it down properly.

Ok, I now see that you were only trying to clarify my notation. I would agree that your notation emphasizes that hyper-iterations are actually: function * OUTPUT of a function -> function.

tongue in cheek edit: all this editing posts only to see it was too late biz, i know... wastes such a h**k of time...