I am having trouble understanding the following lemma in Kanamori's book. Some notations: $\mathcal{M}(T,\alpha)$ denotes the (unique up to isomorphism) model of $T$ generated by a set of indiscernibles of order type $\alpha$. Here since $T$ preserves well-foundedness $\mathcal{M}(T,\alpha)$ is already identified with some unique $L_\iota$.
Why is the Skolem hull $\mathcal{N}$ literally equal to $\mathcal{M}(T,\alpha)$? I understand there is only one transitive model isomorphic to $\mathcal{M}(T,\alpha)$, but I cannot see why is $\mathcal{N}$ transitive. From the proof I am somewhat convinced that the transitive collapse of $\mathcal{N}$ has the same height as $\mathcal{N}$, and is equal to $\mathcal{M}(T,\alpha)$, but from the later text it seems we are indeed talking about $\mathcal{N}$ rather than its collapse. So what is the reasoning here?
Also why is $\iota^{T,\alpha}_\xi=\iota^{T,\beta}_\xi$? Is it because they are both indiscernibles generating $\mathcal{M}(T,\alpha)$, and if they were different that would induce a non trivial automorphism of $\mathcal{M}(T,\alpha)$?
A bit of progress: denote the collapse of $\mathcal{N}$ as $\mathcal{N}_0$. The collapse is identity on ordinals. The inverse of the collapsing map is an elementary embedding of $\mathcal{N}_0$ into $\mathcal{M}(T,\alpha)$. It is identity on ordinals. Therefore it seems very plausible to me that it is inclusion, which means the collapse is identity. There are two issues, first I do not know if these models are models of $ZFC$ (they should be; when defining the EM blueprint the author uses the least $\rho$ s.t. $L_{\rho}$ contains an uncountable set of indiscernibles; I wonder if this is the same as the first Erdos cardinal $\kappa(\omega_{1})$). Second issue is that to claim it's inclusion I want to appeal to Proposition 5.1 in the same book, but I am not certain if 5.1 implies that. Another random question: is it impossible for the inclusion between two proper classes to be elementary?