  A transfunction is a function which maps between sets of finite measures on
measurable spaces. In this paper we characterize transfunctions that correspond
to Markov operators and to plans; such a transfunction will contain the
"instructions" common to several Markov operators and plans. We also define the
adjoint of transfunctions in two settings and provide conditions for existence
of adjoints. Finally, we develop approximations of identity in each setting and
use them to approximate weakly-continuous transfunctions with simple
transfunctions; one of these results can be applied to some optimal transport
problems to approximate the optimal cost with simple Markov transfunctions.
