Let $\mathcal S$ be a base category and let $Fib_\mathcal S^{split}$ be the 2-category consisting of split fibrations, fibered functors which strictly preserve the splitting and vertical transfomations between them. Let $U:Fib_\mathcal S^{split} \to Fib_\mathcal S$ be the 2-functor into the 2-category of fibrations which forgets the splitting. This 2-functor is not a hom-cat-wise equivalence, but it is (according to Streicher - Fibered Categories - page 13) the 2-localisation of $Fib_\mathcal S^{split}$ at the class of 1-cells which become equivalences after applying $U$.
Often when one tries to define the interpretation of a type-constructor in the category $\mathcal S$ one has to solve the following problem. One has some 1-cell $F:\mathbb A\to \mathbb B$ in $Fib_\mathcal S^{split}$ and to interpret the type-constructor one needs an (right or left) adjoint to that 1-cell in $Fib_\mathcal S^{split}$. Often the categorical properties of the underlying category $\mathcal S$ will ensure that the corresponding 1-cell $U(F)$ has an adjoint in $Fib_\mathcal S$, but since $U$ is not hom-cat-wise an equivalence one can not lift that adjoint to a strict one. In consequence, one has to show in a case by case manner that one can construct a split adjoint, and this is always extremly tedious.
Since $U$ is a localisation, I wanted to ask if there are some general (model theoretic? homotopical?) methods or shorthands which one can use to show that in some cases an adjoint of $U(F)$ in $Fib_\mathcal S$ lifts to one in $Fib_\mathcal S^{split}$?
I apologize if the question is to vague to have a satisfying answer. It would take some time to write down an example, since the definitions are usually a bit involved, but I could do that if it is helpful.
