Let $F:C \to D$ be a functor. Then, for all $c \in C_0$ we have an induced functor $F/c:C/c \to D/Fc$. Suppose that for each $c$, $F/c$ has a right adjoint. There's surely a name for such a functor. What is it and where can I read about them?
Note: I am not assuming that $C$ has a terminal object- in fact, the example I have in mind does NOT have one.