Let $\Lambda$ be the Iwasawa Algebra of the Galois group of the cyclotomic $\mathbb{Z}_p$-extension $\mathbb{Q}_{cyc}$ of $\mathbb{Q}$. Let $\widehat{\Lambda}$ be its Pontryagin dual (i.e the dualizing module is $\mathbb{Q}_p/\mathbb{Z}_p$). Let $\tilde{\kappa}$ be the "universal $p$-adic cyclotomic character" $G_\mathbb{Q} \rightarrow Gal(\mathbb{Q}_{cyc}/Q) \rightarrow \Lambda^\times$.

Let $\rho: G_\mathbb{Q}\rightarrow Aut_{\mathbb{Z}_p}(T_p(E))$ be the Galois representation attached to the Tate module of an elliptic curve over $\mathbb{Q}$. Let $\tilde{\rho}: G_\mathbb{Q} \rightarrow Aut_{\Lambda}(\tilde{T}_p(E))$ be the cyclotomic deformation, where $\tilde{T}_p(E)= T_p(E) \otimes_{\mathbb{Z}_p} \Lambda(\tilde{\kappa})$. Here $\Lambda(\tilde{\kappa})$ is the $\Lambda$ but the Galois action is given by $\tilde{\kappa}$.

Let $\widehat{\Lambda}(\tilde{\kappa})$ denote $\widehat{\Lambda}$ where $g \in G_\mathbb{Q}$ acts by $\tilde{\kappa}(g)$. Then we know (by Greenberg) that $$\widehat{\Lambda}(\tilde{\kappa})=Hom(\widehat{\Lambda}(\tilde{\kappa}^{-1}), \mathbb{Q_p}/\mathbb{Z}_p).$$

What I am trying to understand is the following.

Greenberg says that $$T_p(E) \otimes_{\mathbb{Z}_p}Hom(\widehat{\Lambda}(\tilde{\kappa}^{-1}), \mathbb{Q_p}/\mathbb{Z}_p) \cong Hom(\widehat{\Lambda}(\tilde{\kappa}^{-1}), T_p(E) \otimes_{\mathbb{Z}_p}\mathbb{Q_p}/\mathbb{Z}_p) $$

I know from this math overflow post Does module Hom commute with tensor product in the second variable? that to have $Hom_A(L,M) \otimes_A N \cong Hom_A(L, M \otimes_A N)$ we need $L$ to be finitely presented and $N$ to be flat.

But in Greenberg's situation, $\widehat{\Lambda}(\tilde{\kappa}^{-1})$ does not seem to be finitely generated $\mathbb{Z}_p$-module. This is the problem I am facing. Any help is appreciated.

Greenberg's statement is attached. It is an extract of his paper "Iwasawa theory and p-adic deformation of motives" (page 214). See the red highlighted part.