In Streicher's text "Fibered categories" it is described how to construct a 2-functor $Sp:Fib_\mathscr S\to 2Cat_s(\mathscr S^{op},Cat)$ such that $R = Sp\circ \int$ is right 2-adjoint to the forgetful 2-functor $U:2Cat_s(\mathscr S^{op},Cat)\to 2Cat(\mathscr S^{op},Cat)$. The first 2-category is here the 2-category of strict 2-functors, strict transformations and modifications. The second 2-category is the 2-category of pseudofunctors, pseudo-natural transformations and modifications. The 2-functor $\int$ is the Grothendieck construction $\int: 2Cat(\mathscr S^{op},Cat) \to Fib_\mathscr S$. Streicher's text only claims that there is a 2-natural equivalence $$2Cat_s(A,RB)\simeq 2Cat(UA,B)$$ Is it true that this can be upgraded to a $Cat$-enriched adjunction, i.e. that the equivalence above is in fact an isomorphism of categories which varies naturally?
I am asking because on page 46 example 8.23 second paragraph in Shulman's paper "All $(\infty,1)$-toposes haves strict univalent universes" it is claimed that $RUC$ strictly classifies pseudomorphism, i.e. that a map $UA\to UC$ is exactly the same thing as a strict transformation $A\to RUC$. This sounds like we should have an isomorphism and not just an equivalence in the adjunction above.
I think I have checked on pen-and-paper that there is an isomorphism, but 2-categories are a mess, and so I want to make extra sure that I am not mistaken. A reference would also be very, very welcome!