Let $\mathfrak{A}$ be a full subcategory of $\mathfrak{B}$, where $\mathfrak{A}$ and $\mathfrak{B}$ are both concrete over a given category $\mathfrak{C}$. This means that we can consider the two forgetful functors $U_\mathfrak{B}: \mathfrak{B} \to \mathfrak{C}$ and its corresponding restriction $U_\mathfrak{A}$ to $\mathfrak{A}$.
Assume moreover that the previous forgetful functors are also left adjoint to two functors $G_\mathfrak{B}$ and $G_\mathfrak{A}$, respectively. Hence the adjoint pairs $(U_\mathfrak{A},G_\mathfrak{A})$ and $(U_\mathfrak{B},G_\mathfrak{B})$ induce two (different?) monads $T:=T_\mathfrak{B}$ and $E:=T_\mathfrak{A}$ and, in particular, two (different?) Eilenberg-Moore categories $\mathcal{C}^E$ and $\mathcal{C}^T$ and two (different?) Kleisli categories $(\mathcal{C}_E)$ and $(\mathcal{C}_T)$.
Well my question is the following: is there some well-known relationship between $\mathcal{C}^E$ and $\mathcal{C}^T$ and between $(\mathcal{C}_E)$ and $(\mathcal{C}_T)$?