I report our definition of a branching process.
Let $X$ be a random variable with $\mathbb{P}[X=j]=p_j$ and $(X_{n,i})_{n,i\geq 1}$ i.i.d. random variables with $X_{n,i}\overset{\mathcal{L}}{=}X$. Then we define the branching process $Z_n$ to be the number of individuals in the n-th generation, where, by convention, we let $Z_0 = 1$. Then $Z_n$ satisfies the recursion relation $$Z_n=\sum_{i=1}^{Z_{n-1}}{X_{n,i}}.$$
We had a theorem with proof beginning as follows.
Define $\eta_n:=\mathbb{P}[Z_n=0]$, and because $\{Z_n=0\}\subseteq \{Z_{n+1}\}$ we have that $\eta_n\uparrow\eta$. Define also $G_n(s):=\mathbb{E}[s^{Z_n}]$ (generating function of the number of individuals in the n-th generation), so that $\eta_n=G_n(0)=\mathbb{P}[Z_n=0]$. So we condition on the event $\{Z_1=i\}$:
$$G_n(s)=\mathbb{E}[s^{Z_n}]=\sum_{i\geq 0}{p_i\mathbb{E}E[s^{Z_n}\mid Z_1=i]}=\sum_{i\geq 0}{p_i\mathbb{E}[s^{Z_n}\mid Z_1=i]}\overset{*}{=}\sum_{i\geq 0}{p_i\mathbb{E}[s^{Z_{n-1}}]^i}=\sum_{i\geq 0}{p_iG_{n-1}(s)^i}=G_X(G_{n-1}(s))$$
Now my problem is $(*)$. I understand the reasoning made here. Instead to consider the BP starting from $Z_1=i$ till $Z_n$ we can consider $i$ processes starting from $Z_0=1$. But it's not clear how to formally write it down.