Strongly involutory functors

Let C be an arbitrary category. We study strongly involutory functors on C, defined as involutory contravariant endofunctors of C acting as identity on objects. Motivating examples can be constructed if we think at the transpose of a matrix, the adjoint of a linear continuous operator between two Hilbert spaces, and the inverse of a morphism in a groupoid. We show how a strongly involutory functor on a skeleton of C extends to C, and we apply this to find all such functors for a groupoid. We describe and classify up to a natural equivalence all strongly involutory functors on the category of finite dimensional vector spaces over a field. Strongly involutory functors with a special property related to generalized inverses of morphisms are studied.