Troubles with Kanamori's Theorem 7.8 In Kanamori's Book The Higher Infinite is proved in Theorem 7.8. (implication (b) $\rightarrow$ (a) ) that every inaccesible cardinal with the tree property is a weakly compact cardinal by means of the extension property.
The idea of the proof is bild up a $\kappa-$tree (which will have a branch of size $\kappa$), consider the direct limit of one of its branches and finally take the transitive collapse of that limit of models. I'm having troubles understanding how Kanamori is able to ensure that the cardinal $\kappa$ lies in that transitive collpase $\langle X,\in, R\rangle $. He says that every $\beta_\xi$ collapse to the same ordinal $\beta\in X$ and thus this $\beta$ must be greater or equal than $\kappa$. I adjoint a screenshot of the proof:

I don't see why every ordinal $\beta_\xi$ collapses to the same ordinal and moreover how it implies that $\beta\geq \kappa$. Could someone explain me the details of this last part? 
Best,
Cesare.
 A: Let $( [H(\xi, \beta_\xi)] \mid \xi < \beta)$ be a cofinal branch in $T$ and let $M$ be the direct limit of the linear system $( H(\xi, \beta_\xi), i_{\xi, \eta} \mid \xi \le \eta < \kappa)$, where $i_{\xi, \eta} \colon H(\xi, \beta_\xi ) \to H(\eta, \beta_\eta)$ is an elementary embedding such that $i_{\xi,\eta} \restriction V_{\alpha_\xi} = \operatorname{id}$ and $i_{\xi, \eta}(\beta_\xi) = \beta_\eta$.
We identify $M$ with its transitive collapse. For each $\xi < \kappa$ let $\pi_\xi \colon H(\xi, \beta_\xi) \to M$ be the elementary embedding into the direct limit.
We first claim that there is an ordinal $\beta$ such that $\pi_\xi(\beta_\xi) = \beta$ for all $\xi < \kappa$:
Proof. Since $H(\xi, \beta_\xi) \models \beta_\xi \text{ is an ordinal}$ and $\pi_\xi$ is elementary, we have that $M \models \pi_\xi(\beta_\xi) \text{ is an ordinal}$. Now $M$ is transitive and 'being an ordinal' is a $\Sigma_0$-property. Thus $\beta := \pi_\xi(\beta_\xi)$ is an actual ordinal. 
We claim that $\beta$ is independent of $\xi < \kappa$: Let $\xi \le \eta < \kappa$. Since $M$ is a direct limit we have $\beta = \pi_\xi(\beta_\xi) = \pi_\eta \circ i_{\xi, \eta}(\beta_xi) =  \pi_\eta( \beta_\eta)$. Q.E.D.
Next, we claim that $\beta \ge \kappa$:
Proof. Assume, toward a contradiction, that $\beta < \kappa$. Since $( [H(\xi, \beta_\xi)] \mid \xi < \beta)$ is cofinal in $T$, there is some $\xi < \kappa$ such that $\beta < \alpha_\xi$. Since $H(\xi, \beta_\xi) \cap V_{\alpha_\xi}$ is transitive, we have $\beta \in H(\xi, \beta_\xi)$. By the elementarity of $\pi_\xi$ we now have $\pi_\xi(\beta) < \pi_\xi(\beta_\xi) = \beta$. This is impossible, because for any homomorphism
$$
\rho \colon (N;\in) \to (N'; \in)
$$
between transitive structures and any ordinal $\gamma \in N$, we always have $\rho(\gamma) \ge \gamma$. (Otherwise let $\delta$ be the least ordinal such that $\rho(\delta) < \delta$. Then $\rho(\delta) \in N$ and, by because $\rho$ is a $\in$-homomorphism, $\rho(\rho(\delta)) < \rho(\delta) < \delta$. Thus $\delta' := \rho(\delta)$ satisfies $\rho(\delta') < \delta'$ - contradicting the choice of $\delta$.) Q.E.D.
