The question was originally written in German, but from what I understand, the gist is, that I'm supposed to show that the mapping is both left-total and right-unique.
Let A be a set, $a \in A$ and $h: A \to A$ a mapping.
$f$ is defined recursively:
(i) $f(0) = a $
(ii) $f(n+1) = h(f(n))$ for all $n \in \mathbb{N}$
Through this a subset of $N \times A$ is defined, this is however not obvious.
Prove:
(a) There is a maximum of one mapping $f: N \to A$ with the properties (i) and (ii).
(b) There is one mapping $f: N \to A$ with the properties (i) and (ii).
So far what I think is the case is that (a) is related to mapping being right-unique:
$ \forall n \in \mathbb{N} $ and $ \forall a_1, a_2 \in A, nRa_1 \wedge nRa_2 \implies a_1 = a_2$
And I believe that (b) is left-total:
$\forall n \in \mathbb{N}, \exists a \in A$ so that $nRa$
I noticed that the mapping looks like a composition of functions e.g. $n = 3$ you get: $f(3) = h(h(h(a)))$ i.e. $ h \circ h \circ h$
I'd really appreciate if someone could give me some advice on how to proceed!