For a field extension $L/k$, there is a base change functor $\textbf{Sch}/k \rightarrow \textbf{Sch}/L$. The answer to the MO question Why is the base change functor faithful has shown that this functor is always faithful.
Question 1: Is this functor always full? If not, could anyone provide a counter-example?
Question 2: If $L/k$ is a finite separable extension, in this nice case, is the base change functor fully faithful?