  We prove that cloven Grothendieck fibrations over a fixed base $\ct{B}$ are
the pseudo-coalgebras for a lax idempotent 2-comonad on $\ct{Cat}/\ct{B}$. We
show this via an original observation that the known colax idempotent 2-monad
for fibrations over a fixed base has a right 2-adjoint. As an important
consequence, we obtain an original cofree construction of a fibration on a
functor. We also give a new, conceptual proof of the fact that the forgetful
2-functor from split fibrations to cloven fibrations over a fixed base has both
a left 2-adjoint and a right 2-adjoint, in terms of coherence phenomena of
strictification of pseudo-(co)algebras. The 2-monad for fibrations yields the
left splitting and the 2-comonad yields the right splitting. Moreover, we show
that the constructions induced by these coherence theorems recover Giraud's
explicit constructions of the left and the right splittings.
