For a covariant functor W. Fulton and R. MacPherson defined an operational bivariant theory associated to this covariant functor. In this paper we will show that given a contravariant functor one can similarly construct a “dual” version of an operational bivariant theory, which we call a co-operational bivariant theory. If a given contravariant functor is the usual cohomology theory, then our co-operational bivariant group for the identity map consists of what are usually called “cohomology operations”. In this sense, our co-operational bivariant theory consists of “generalized” cohomology operations.