Let $\mathcal{E}$ be the ring of series $f(T)=\sum_{n \in \mathbb{Z}} a_nT^n, \; a_n \in \mathbb{Q}_p$ such that the $\{a_n\}$ are bounded and $a_{-n}\rightarrow 0$ as $n \rightarrow +\infty$. This is the usual coefficient ring in Fontaine's theory of $(\varphi, \Gamma)$-modules. This ring comes equipped with a Frobenius endomorphism $\varphi$ and a derivation $\frac{d}{dT}$. I recently read that any finite free module over $\mathcal{E}$ equipped with a unit-root $\varphi$-structure automatically admits a unique compatible differential modules structure. Why is this true?

Conversely, if we take a finite free module $M$ over $\mathcal{E}$ equipped with a differential module structure can we find a compatible unit-root Frobenius structure on $M$? Furthermore, if we can find one, to what extent is it unique?

Let $\mathcal{E}$ be the ring of series $f(T)=\sum_{n \in \mathbb{Z}} a_nT^n$, $a_n \in \mathbb{Q}_p$ such that the $\{a_n\}$ are bounded and $a_{-n}\rightarrow 0$ as $n \rightarrow +\infty$. This is the usual coefficient ring in Fontaine's theory of $(\varphi, \Gamma)$-modules. This ring comes equipped with a Frobenius endomorphism $\varphi$ and a derivation $\frac{d}{dT}$. I recently read that any finite free module over $\mathcal{E}$ equipped with a unit-root $\varphi$-structure automatically admits a unique compatible differential module structure. Why is this true?

Conversely, if we take a finite free module $M$ over $\mathcal{E}$ equipped with a differential module structure can we find a compatible unit-root Frobenius structure on $M$? Furthermore, if we can find one, to what extent is it unique?