Re: local maps of toposes are always UIAO

Thomas Streicher asked

I'd like to know whether the following simple observation is well known. If F -| U : E -> S is a local map of toposes i.e. Gamma : Gl(F) -> Gl(Id_S) has a fibred right adjoint Nabla then U is full and faithful, i.e. one has the situation of a Unity and Identity of Adjoint Opposites in Lawvere's sense.

Yes, there is a simple proof of this fact in Proposition 1.4 of "Local maps of toposes" by Johnstone & Moerdijk (Proc. London Math. Soc. (3) 58 (1989), 281--305).