The category of finite-dimensional Banach spaces and
linear contractions is a symmetric star-autonomous
category which is not compact closed; in other words,
"the" tensor product is not self-dual; which is to say,
there are really two monoidal structures, the "usual"
one (which is closed) and its dual (which is co-closed),
linked by linear distributions; but they have the same
unit, so they are also linked by a mix map.

The category of all Banach spaces and linear
contractions is not star-autonomous, but both of the
monoidal structures extend to the larger category, and
continue to be linked by linear distributions and mix
maps. I will try to explain why as abstractly as
possible, using only the fact that Ban is a closed
monoidal structure with a well-behaved factorisation
system. (In particular, I will not use the fact that
Ban_fd is star-autonomous.)

In principle, this means that similar mix structures
might be induced on other categories, and indeed I do
know of some, though none outside of functional
analysis.