Confusion about the rephrase of Recursion Theorem

Question

From textbook A Course in Mathematical Analysis by Prof D. J. H. Garling, I'm confused about how he rephrases the Recursion Theorem.

First, he states the theorem:

Then he says:

Finally, he expresses the theorem in a more general term:

My question is: the author says "there exists a unique mapping $f^{n}: A → A$", but I feel like there are more than one mappings: $f^{0} : A → A, f^{1} : A → A, f^{2} : A → A,...$.

Many thanks for clarifying my doubt!

Answer 1

What you feel is exactly what the theorem says! It does NOT say that "there exists a unique mapping $f^n$ …". It says that

For each $\color{blue}{n}\in\mathbb{Z}^{+}$ there exists a unique mapping $f^{\color{blue}{n}}:A\to A$ …

So all in all there are many these mappings, just as you said: there's one for $n=0$, one for $n=1$, one for $n=2$, etc.