Abstract
Consider categories with involutions which fix objects, functors which preserve involution, and natural transformations. In this setting certain natural adjunctions become universal and, thereby, become constructible from abstract data. Although the formal theory of monads fails to apply and the Eilenberg-Moore category fails to fit, both are successfully adapted to this setting, which is a 2-category. In this 2-category, each monad (= triple = standard construction) defined by an adjunction is characterized by a pair of special equations. Special monads have universal adjunctions which realize them and have both underlying Frobenius monads and adjoint monads. Examples of monads which do (respectively, do not) satisfy the special equations arise from finite monoids (= semigroups with unit) which are (respectively, are not) groups acting on the category of linear transformations between finite dimensional Euclidean (= positive definite inner product) spaces over the real numbers. More general situations are exposed.
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