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 ?