Is there a standard method for showing that a functor $F:\mathcal{C}\to\mathcal{D}$ is a fibration, aside from constructing a cleavage?
In the proof of the Grothendieck construction, the fibration we obtain from an indexed category $\Psi:\mathcal{B}^{op}\to\mathfrak{Cat}$ is automatically cloven since we're constructing a specific Cartesian arrow $(u,1_{\Psi(u)(Y)})$ for each arrow $u:I\to J\in\mathcal{B}$ and object $(J,Y)\in\int\Psi$ above $J$.
Every time I want to show that a functor is a fibration, I end up constructing Cartesian arrows parametrized as above and thusly showing that it's a cloven fibration -- is this by necessity?
Any method of showing that a functor is a fibration without choosing a cleavage is welcome, but in particular something similar to the adjoint functor theorem for fibrations would be cool. That is, a statement along the lines of
If $F:\mathcal{C}\to\mathcal{D}$ is a functor and $\mathcal{C}$ is blah and $\mathcal{D}$ is bloop and $F$ preserves/reflects blorps then $F$ is a fibration.