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(−)×F(−) → F(−×−) 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.