TITLE: Fibered category with an adjoint inclusion
QUESTION [6 upvotes]: Suppose $X:D \to C$ is a fibered category (I do not assume the fibers to be groupoids). Suppose that $X$ is actually left adjoint to a fully faithful embedding $C \hookrightarrow D$. Is there a special name for such a fibration $X$?

REPLY [3 votes]: Isn't the answer simply that the fibration has fibred terminal objects?