A monad of product type \(M = \Pi \Delta\).
Composing the functor \(\Delta :: C \to C \times C\) (which takes
objects to pairs of objects) and it's right adjoint the product
functor, \(\Pi :: C \times C \to C\), gives a monad \(M\) when
equipped with natural transformations \(\eta :: Id \to M \) and
\(\mu :: M^2 \to M \).
The unit, η, acts to include the dynamical system along the
diagonal in M,
while the join, 𝜇, acts to fold \(M^2\)
onto its factor \(M\).
Above are two dynamical systems. Thick black borders around a
state indicate a fixed point of the system with all other
states 'flowing' down to this one. Let the large red dots on the
left represent the dynamical system, \(d :: Set \to Set\).
The system on the right is then the result of applying \(M\) to
\(d\). Note the small red dots on the left correspond to those
on the right. This is to highlight the
inclusion along the diagonal of \(d\) via \(\eta :: Id \to M\).