think of $F$ identifying the old identity arrow at an object $x$ with the empty string at $x$.

This adjoint pair gives a comonad on $Cat$ in the usual way, and hence a functorial simplicial resolution, which we will denote $S(\mathbb{A})\to \mathbb{A}$. In more detail, we write $T = FU$ for the functor part of the comonad, the unit of the adjunction $\eta : Id_{DGrph_0} \to UF$ gives the comultiplication $F\eta U: T \to T^2$ and the counit of the adjuction gives $\varepsilon : FU \to Id_{Cat}$, that is, $\varepsilon : T \to Id$. Now for $\mathbb{A}$ a small category, set $S(\mathbb{A})_n = T^{n+1}(\mathbb{A})$ with face maps $d_i : T^{n+1}(\mathbb{A}) \to T^n(\mathbb{A})$ given by $d_i = T^ {n-i}\varepsilon T^i$, and similarly for the degeneracies which use the comultiplication in an analogous formula. (Note that there are two conventions possible here. The other will use $d_i = T^i\varepsilon T^{n-i}$. The only effect of such a change is to reverse the direction of certain ‘arrows’ in diagrams later on. The two simplicial structures are ‘dual’ to each other.)

This $S(\mathbb{A})$ is a simplicial object in $Cat$, $S(\mathbb{A}) : \mathbf{\Delta}^{op} \to Cat$, so does not immediately gives us a simplicially enriched category, however 