TITLE: If $F:C\to D$ and $G:D\to E$ are functors, and both $GF$ and $G$ have a right adjoint, does $F$ has a right adjoint too?
QUESTION [6 upvotes]: Suppose that $F:C\to D$ and $G:D\to E$ are functors such that both of $GF$ and $G$ have a right adjoint. Is it true that also $F$ has a right adjoint? And what if only $GF$ has a right adjoint?

REPLY [4 votes]: Let $E$ be a terminal category. Then a right adjoint of a functor $B\to E$ exists if, and only if, $B$ has a terminal object. So, to manufacture a counter example, all you need is to find two categories, $C,D$, and a functor $F:C\to D$, such that

$C$ and $D$ have a terminal object
$F$ has no right adjoint. 

So, take $C=Grp$ and $D= Set$, and $F:C\to D$ to be the forgetful functor. There are of course many other examples, showing that in general adjunctions simply don't behave quite so nicely.