In the categorical interpretation, ! is characterized as the right adjoint to the inclusion between closed monoidal category C and the category of comonoids in C, denoted ComC. So, in the formulation (II), B is supposed to be an object of ComC while A is of C. Then the answer to the question is they are equivalent when the right adjoint ! is onto. In other words, B |- B\otimes B is derivable only when B = !C for some object C. Masaru Shirahata