$\require{AMScd}$I am in the following situation: the diagram $$ \begin{CD} \cal M @>r>> [{\cal B},Set] \\ @VuVV @VVf^*V \\ \cal D @>>N_g> [{\cal A}^\text{op},Set] \end{CD} $$ is a (strict) pullback in $\bf Cat$; moreover, $f : \cal A^\text{op}\to B$ is bijective on objects (and $f^*$ is the "inverse image" functor given by precomposition). In turn, $g$ is the "nerve" induced by a functor $g : \cal A \to D$. $N_g$ is fully faithful because $g$ happens to be dense, and so $r$ is f.f. as well. If needed, $\cal D$ is complete and cocomplete.

This gives $\cal M$ a very explicit description: it is a reflective subcategory of $[{\cal B},Set]$ made by the functors $F : {\cal B}\to Set$ such that $F(fA)={\cal D}(gA,D)$ for some $D\in\cal D$.

Is all this sufficient to imply that $u$ is monadic? If not, what additional assumptions are needed?