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Re: Question about !



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