We give a brief summary of the Horn variety, quasivariety and Birkhoff variety theorems (using the H, S, P and P+ closure operators) for categories of algebras for an endofunctor and dualize these results to yield formally dual theorems for categories of coalgebras. While the notions of coequations and conditional coequations have been covered in some detail previously, this is the first dualization of sets of so-called Horn equations (which include negations of equations). We provide a few natural examples of "Horn covarieties" and close with a brief discussion of "cofree-for-V" coalgebras.