My questions stems from my attempt to understand the paper of Greenberg and Stevens about the Mazur-tate-Teitelbaum conjecture (you can find the paper here). To understand this question you probably need to be familiar with this paper.
Let $f$ be a weight two newform of conductor $Np$ which is split-multiplicative at $p$. Let $K_f$ be the completion of the field of eigenvalues of $f$ and $V_f$ be the two dimensional galois representation over $K_f$ associated to $f$. We will denote by $V_\infty$ the $\Lambda$-adic galois representation associated to the Hida family passing through $f$.
Let $\mathcal{R}$ be Hida's universal ordinary hecke algebra of tame level $N$ and $(\kappa)$ be the ideal of $\mathcal{R}$ corresponding to $f$ (i.e. the kernel of the morphism $\lambda_f : \mathcal{R} \to \overline{\mathbf{Q}_p}$ given by $Tf = \lambda_f(T)f$)
I would like to understand why the short exact sequence (3.19) page 425 : $$ 0 \to \mathcal{R}_{(\kappa)}(\chi_0\eta\varphi^{-1}) \to V_\infty \otimes_\mathcal{R} \mathcal{R}_{(\kappa)} \to \mathcal{R}_{(\kappa)}(\varphi) \to 0$$ specializes modulo $(\kappa)$ to $$0 \to V_f^0(1) \to V_f \to V_f^{et} \to 0$$
I know why the middle term specializes to $V_f$ (this is the whole point of $V_\infty$) but not the other two.
