smn-theorem: Application by instantiating s with other function The smn-Theorem on the basis of Turing Machines and computable functions rather than programs, as in the Wikipedia article for instance, can be defined as follows:

$$
\begin{align*}
&amp;&amp;&amp; s \langle m, n \rangle &amp;&amp;= \lceil M' \rceil \\
&amp;&amp;\implies &amp;\varphi_{s\langle m, n \rangle} &amp;&amp;= \varphi_{\lceil M' \rceil}\\
&amp;&amp;&amp;&amp;&amp;= f_{M'}\\
&amp;&amp;\implies &amp;\varphi_{s\langle m, n \rangle} (i) &amp;&amp;= f_{M'}(i)\\
&amp;&amp;&amp;&amp;&amp; = f_{M_m} \circ f_{M_{\langle n, . \rangle} (i)}\\
&amp;&amp;&amp;&amp;&amp;=\varphi_m \langle n, i\rangle
\end{align*}$$

where $M_{\langle n, . \rangle}$ denotes, that machine $M_{\langle \rangle}$ has $n$ already pre-written on its tape and $\lceil M \rceil$ denotes the Gödel-number of machine $M$.

Now, I am confronted with the claim, that  

$$ \varphi_{h(n)}(i) = f\langle n, i \rangle $$

because

$$ f := \varphi_m \in \text{set of computable functions}$$
then
$$f\langle n, i \rangle = \varphi_m \langle n, i \rangle = \varphi_{s \langle m, n \rangle}(i)$$

therefore it has to be that

$$ h(n) = s\langle m, n \rangle $$

so I guess then it has to be like this:
$$h(n) = \lceil M' \rceil = \lceil f_{M_m} \circ f_{M_{\langle n, . \rangle }} \rceil = s\langle m, n \rangle$$

And so $h$ contains $f_{M_m}$ ... basically just because there can be such a function? Am I interpreting this right?
I think you are reading it backwards.

Usually you want to find $h$ such that $\varphi_{h(n)}(i) = f(n,i)$, where $f$ is a computable partial function. Further, you want $h$ to be total recursive (or even more: primitive recursive).

In order to prove that such an $h$ exists, as well as to concretely define it, one picks any index $m$ for $f$, so that $\varphi_m = f$, and chooses $h(n) = s(m,n)$.

Note that, when reading the reasoning above in the other direction, it becomes wrong. That is: it is not the case that, since $\varphi_{h(n)}(i) = f(n,i)$ we can conclude $h(n) = s(m,n)$.

Indeed, $h$ may also be another function, like e.g. $h(n) = pad(s(m,n))$ where $pad$ is a padding function (which returns the Gödel number for an equivalent TM, such that said Gödel number is larger). Infinitely other (computable) $h$ exist.