Liftings of normal functors in the category of compacta to categories of topological algebra and analysis

We prove that the liftings of a normal functor $F$ in the category of compact Hausdorff spaces to the categories of (abelian) compact semigroups (monoids) are determined by natural transformations $F(-)\times F(-)\to F(-\times-)$ satisfying requirements that correspond to associativity, commutativity, and the existence of a unity. In particular, the tensor products for normal monads satisfy (not necessarily all) these requirements.
It is proved that the power functor in the category of compacta is the only normal functor that admits a natural lifting to the category of convex compacta and their continuous affine mappings.