This is a cross-post from here

I am reading A. Mazel-Gee's paper "All about the Grothendieck construction". In that paper he explains that the left adjoint ${\mathrm{Cat}_{\infty}}_{/\mathcal{C}}\to \mathrm{coCFib}(\mathcal{C})$ (from $\infty$-categories over $\mathcal{C}$ to cocartesian fibrations over $\mathcal{C}$) to the forgetful functor is the functor that sends $F:\mathcal{D}\to \mathcal{C}$ to the "free cocartesian fibration on F" $$\mathrm{Fun}([1],\mathcal{C})\times_{\mathcal{C}}\mathcal{D}\to\mathcal{C}$$ I am now wondering if there is a similar explicit description for the left adjoint ${\mathrm{Cat}_{\infty}}_{/\mathcal{C}}\to \mathrm{LFib}(\mathcal{C})$. This would be the composite of the previous functor with the reflexive localization $L:\mathrm{coCFib}(\mathcal{C})\to \mathrm{LFib}(\mathcal{C})$. Now by the results in the paper we have a commutative diagram of large $\infty$-categories $$\require{AMScd}\begin{CD}\mathrm{Fun}(\mathcal{C},\mathrm{Cat}_\infty) @>{(=)^{gpd}\circ -}>> \mathrm{Fun}(\mathcal{C},\mathcal{S})\\ @V{Gr}V{\simeq}V @V{Gr}V{\simeq}V \\ \mathrm{coCFib}(\mathcal{C}) @>{L}>> \mathrm{LFib}(\mathcal{C}),\end{CD}$$ where $\mathcal{S}$ is the $\infty$-category of spaces, $Gr$ denotes the Grothendieck construction and $(=)^{gpd}$ is the groupoidification functor.

This implies by the naturality of the Grothendieck construction that the fibers of $L(\mathcal{D}\to\mathcal{C})$ over $x$ identify with $(\mathcal{D}_x)^{gpd}$. But it is not straight-up groupoidification as that would take us to $\mathcal{S}_{/\mathcal{C}^{gpd}}$. If I understand correctly the description of the Grothendieck construction as a lax colimit then the functor L should be some kind of "free groupoidification of the fibers". But this is not as explicit as I would like : can we describe this process without referring to the functor by which the coCartesian fibration is classified ?

On the level of model categories, this is presented by the Quillen adjunction $${\mathrm{Set}_{\Delta}^+}_{/\mathcal{C}^\sharp} \leftrightarrows {\mathrm{Set}_\Delta}_{/\mathcal{C}}$$ between the functor forgetting the marked edges and the functor marking all edges ; the model structures are the marked one and the covariant one, respectively. Therefore the functor $L$ is given by a fibrant replacement of a coCartesian fibration $\mathcal{D} \to \mathcal{C}$ in ${\mathrm{Set}_\Delta}_{/\mathcal{C}}$. Do we have explicit such replacements ?