If M and N are modules, the concept of semiregularity (and regularity) of hom (M, N) is defined and studied, and the connection with the relative direct injective- and direct projective- properties is established. The relationship of semiregularity to the Jacobson radical of hom(M,N), to the singular and cosigular ideals of hom(M,N), and to the notion of lying over or under a direct summand , is described, and the basic results in the module case are extended.