Finn is absolutely right that this is a case of doctrinal adjunction, but the situation is complicated by the fact that there are two 2-monads we might be considering. First, there is the 2-monad $T_1$ on the 2-category $K_1=[\mathrm{ob} C, \mathrm{Cat}]$, whose (pseudo)algebras are (pseudo)functors $C^{\mathrm{op}} \to \mathrm{Cat}$. A lax/colax/pseudo morphsim of $T_1$-algebras is precisely a lax/colax/pseudo natural transformation (which, by the way, is one way to remember the correct meanings of lax vs. colax in the latter case).

In this case, doctrinal adjunction tells us that if F and G are (pseudo) $T_1$-algebras and $\Phi\colon F\to G$ is a pseudo $T_1$-morphism, then

If $\Phi$ has a left adjoint in the underlying 2-category $K_1$, then this left adjoint automatically has a structure of colax $T_1$-morphism (this only requires $\Phi$ to be lax), while

If $\Phi$ has a right adjoint in the underlying 2-category $K_1$, then this right adjoint automatically has a structure of lax $T_1$-morphism (this only requires $\Phi$ to be colax).

Note that having an adjoint in $K_1$ is precisely the hypothesis you proposed: that each fiber functor $\Phi_U$ has an adjoint. In general, it is not automatic in either case above that the adjoint $T_1$-morphism is pseudo; the invertibility of its (co)lax structure map is an additional condition to be imposed.

Secondly, there is a different 2-monad $T_2$ on the 2-category $K_2 = \mathrm{Cat}/C$, whose (pseudo)algebras are (cloven) fibrations. This 2-monad is what's called *colax-idempotent*, which means that *every* morphism in $K_2$ between $T_2$-algebras admits a unique structure of colax $T_2$-morphism. It follows (not entirely obviously) that any lax $T_2$-morphism is actually pseudo (its unique colax structure is necessarily an inverse to any lax structure one might give it).

One can check that pseudo $T_2$-morphisms are exactly morphisms of fibrations (i.e. functors over $C$ preserving cartesian arrows), and therefore correspond (under the equivalence between fibrations and pseudofunctors) to pseudo natural transformations, i.e. to pseudo $T_1$-morphisms. Similarly, colax $T_2$-morphisms (that is, arbitrary functors over $C$) correspond to colax natural transformations (colax $T_1$-morphisms); but there is no way to represent *lax* natural transformations between pseudofunctors in terms of their corresponding fibrations.

(As an aside, the passage from $T_1$ to $T_2$ is an instance of a general construction which, given a well-behaved 2-monad, produces a new (co)lax-idempotent one with the same algebras. The key word to look for is generalized multicategory).

Now, doctrinal adjunction applied to $T_2$ says that if $\Phi\colon F\to G$ is a pseudo $T_2$-morphism (that is, a morphism of fibrations), then

If $\Phi$ has a left adjoint in the underlying 2-category $K_2$, then that left adjoint automatically becomes a colax $T_2$-morphism, which is no condition at all since $T_2$ is colax-idempotent.

If $\Phi$ has a right adjoint in the underlying 2-category $K_2$, then that right adjoint automatically becomes a lax $T_2$-morphism, and therefore a pseudo one.

This second case is the only one in which we can deduce that the adjoint is actually a pseudo morphism. Note, though, that the hypothesis that $\Phi$ has an adjoint in $K_2$ is stronger than that it has an adjoint in $K_1$.

So what about james-parson's answer? First of all, although he mentions right adjoints, his argument is actually about left adjoints: a universal arrow $y\to \Phi x$ means that x is the value at y of a left adjoint to $\Phi$ (the universal arrow being the adjunction unit). He assumes a left adjoint in $K_1$, which by the general nonsense above should only allow us to deduce the existence of a *colax* $T_1$-structure on this adjoint. But this is equivalent to a colax $T_2$-structure, which (since $T_2$ is colax-idempotent) is equivalent to just saying that we have an adjoint in $K_2 = \mathrm{Cat}/C$. And in fact, that is exactly what he constructs. In general, this adjoint will not be a *pseudo* $T_2$-morphism, i.e. it will not be a morphism of fibrations.