Let $F$ and $G$ be endofunctors over categories $C$ and $ D$, respectively. Suppose that there is a forgetful functor $C \to D$ that has a left adjoint. Can we infer properties of $F$-coalgebras from those of $G$-algebras, or vice versa, by exploiting the connection between the two categories? Presumably the situation here is too vague. Is there some special case of this scenario in which my question make sense and answerable? What are the relevant literature here?
(The application I had in mind is this: suppose that I have a coalgebraic modal logic $L$ for coalgebras in $D$, and that $L$ as it is does not generalize to other categories than $D$ since it relies on some special properties of $D$. Can I still use $L$ to describe coalgebras in $C$, at least in some limited cases, by using the adjunction?)