As Daniel Schäppi pointed out, this is the same as asking whether the composite of two monadic functors is monadic. The answer, unfortunately, is no.

Consider locally presentable categories. By the classification theorem, every locally presentable category can be embedded as a reflective subcategory of some presheaf topos. It is not hard to see that a fully faithful functor with a left adjoint is automatically monadic. On the other hand, $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ is monadic over $\mathbf{Set}^{\operatorname{ob} \mathcal{A}}$, as one might expect for a multi-sorted algebraic theory. Thus, for any locally presentable category $\mathcal{C}$, there exist a small category $\mathcal{A}$ and monadic functors
$$\mathcal{C} \to [\mathcal{A}^\mathrm{op}, \mathbf{Set}] \to \mathbf{Set}^{\operatorname{ob} \mathcal{A}}$$
but in general the composite $\mathcal{C} \to \mathbf{Set}^{\operatorname{ob} \mathcal{A}}$ is *not* monadic.

Indeed, it can be shown that any category monadic over $\mathbf{Set}^I$ (for some set $I$) must be an effective regular (= Barr-exact) category. But this is not true for a general locally presentable category, such as $\mathbf{Cat}$ or $\mathbf{Poset}$.