  We give a new coalgebraic semantics for intuitionistic modal logic with
$\Box$. In particular, we provide a colagebraic representation of
intuitionistic descriptive modal frames and of intuitonistic modal Kripke
frames based on image-finite posets. This gives a solution to a problem in the
area of coalgebaic logic for these classes of frames, raised explicitly by
Litak (2014) and de Groot and Pattinson (2020). Our key technical tool is a
recent generalization of a construction by Ghilardi, in the form of a right
adjoint to the inclusion of the category of Esakia spaces in the category of
Priestley spaces. As an application of these results, we study bisimulations of
intuitionistic modal frames, describe dual spaces of free modal Heyting
algebras, and provide a path towards a theory of coalgebraic intuitionistic
logics.
