  The theory of Mean Field Game of Controls considers a class of mean field
games where the interaction is through the joint distribution of the state and
control. It is well known that, for standard mean field games, certain
monotonicity condition is crucial to guarantee the uniqueness of mean field
equilibria and then the global wellposedness for master equations. In the
literature, the monotonicity condition could be the Lasry-Lions monotonicity,
the displacement monotonicity, or the anti-monotonicity conditions. In this
paper, we investigate all these three types of monotonicity conditions for Mean
Field Games of Controls and show their propagation along the solutions to the
master equations with common noises. In particular, we extend the displacement
monotonicity to semi-monotonicity, whose propagation result is new even for
standard mean field games. This is the first step towards the global
wellposedness theory for master equations of Mean Field Games of Controls.
