In the p-adic Hodge-Tate theory, what does the $\phi$-function in the definition of $(\phi, N)$-module mean?
I mean what is the definition of $\phi$-function here?
I got it that,
A $\phi-$ module is $K_0=\text{Frac} W(k)$-vector space $D$ with an endomorphism $\phi$ such that, writing $\sigma:K_0 \to K_0$ for Frobenius, the map $\phi$ is called semi-linear. This means that $$ \phi(\lambda x)=\sigma (\lambda) \phi(x)$$, for all $\lambda \in K_0$ and for all $x \in D$.
From the above I know that $\phi$ is an endomorphism.
But what is the domain or codomain of $\phi$ here?