Differential equations and categorical logic

Question

In categorical logic, we usually think of each theory as being category $\mathbf{T}$. A model of $\mathbf{T}$ is then a functor $\mathbf{T} \rightarrow \mathbf{Set}$ satisfying some appropriate conditions. And, we can change $\mathbf{Set}$ to other sufficiently-nice categories to get other notions of model. I was wondering whether or not something similar can be done for differential equations/IVP's/integral equations (etc.); can we think of these as some kind of object such that morphisms out of this object "are" solutions to the problem?