An earlier question that I posed sought to define a category with a set of quantum channels as arrows and the C$^{*}$-algebra that these channels map from and to as the object. So, for example, my channels might be the set of all channels with 2 x 2 Kraus operators mapping some C$^{*}$-algebra to itself. Since the channels are always $d \times d$ square matrices, I'll refer to them as having dimension $d$. So far the answers to that question seem to have indicated that my formulation is correct.

Now suppose we have one such category whose channels have dimension 2 that I'll call **Chan**(2). Suppose we also have a category whose channels have dimension 4 that I'll call **Chan**(4). A tensor product of the channels in **Chan**(2) with themselves produces a channel that is in **Chan**(4), i.e. it is a mapping from **Chan**(2) to **Chan**(4). The tensor product is known to be a functor so this isn't unexpected.

But here's my question: is there a way to define an isomorphism between **Chan**(2) and **Chan**(4)? In other words, can the action of the tensor product be "undone," i.e. if the tensor product is the functor going from **Chan**(2) to **Chan**(4), is there a functor going from **Chan**(4) to **Chan**(2) and, can the two together define an isomorphism? If I make a larger category out of all these little categories, it seems like I could make the tensor product and anything that undoes it, arrows in the larger category and then I'd have my isomorphism. Can I do this and, if so, how does one undo a tensor product?