Category of elements and Quillen adjunction

Question

Consider the category of elements construction described in https://ncatlab.org/nlab/show/category+of+elements. It induces a left adjoint from $[\mathcal{C},\mathrm{Set}]$ (the category of functors from the small category $\mathcal{C}$ to the category of sets) to the category of small categories.

Can this left adjoint be "upgraded" as a left Quillen adjoint somehow ?

Answer 1

One can do even better than a Quillen adjunction: Theorem 3.8 in A model structure for Grothendieck fibrations establishes a Quillen equivalence between the projective model structure on presheaves of sets on a small category $C$ and the slice category $\def\Cat{{\sf Cat}}\Cat/C$ equipped with the discrete fibration model structure.