Let $\varphi: A \longrightarrow B$ be a morphism of $k$-algebras (with identity and not necessarily commutative) and $k$ a commutative ring (with identity). Let $F:= {}_B B_A \otimes_A {}_A (-) : \mathbf{Mod}_A \longrightarrow \mathbf{Mod}_B$.
Question) Are there any characterisations of $k$-algebra morphisms $\varphi: A \rightarrow B$ such that $F$ is full?
Thanks in advance.
EDIT As noticed by Jeremy Rickard, my claim that $\varphi: A \rightarrow B$ being fully faithful implied fullness was completely stupid. Indeed, if $F$ is faithful and full, then $\text{End}_A ({}_A A) \cong A^{op} \cong B^{op} \cong \text{End}_B ({}_B B_A \otimes_A A)$. I've modified, then, the question accordingly.