Are there any works on coalgebraic logic (which is, in a way, generalization of modal logic, where we have an object $W$ in a category and a $T$-coalgebra $(W, \gamma)$; modal logic is a spacial case where $W$ is a set and $T = \mathcal{P}$, powerset functor), where the author isn't restricted to category $\mathbf{Set}$ of sets and functions? I've been looking to find some, but I have yet to find any.
Some examples I found were works by Yde Venema, including his chapter in Blackburn, Rijke, Venema: Modal Logic and his lecture notes about coalgebras and modal logic (https://staff.fnwi.uva.nl/y.venema/teaching/ml/notes/cml-20191215.pdf). I also saw some of it in Introduction to Coalgebra by Bart Jacobs, there even is some of it in a more abstractway, but only a few pages.
Could the theory of coalgebraic logic work in a topos? Or even in a general category with some additional structure (terminal object, power objects, etc.)?