fa.functional analysis - Which Banach spaces have categorical duals?

My internet access at the moment is limited & sluggish, so I haven't been able to look up all the details; but I think your reasoning is correct. Certainly my impression is that duals in the sense beloved by (S)MC people only work for finite-dimensional Banach spaces.

By the way, for arbitrary Banach spaces the first map you describe wants to land in the injective tensor product, while the second eants to come out of the projective tensor product. Thus the failure to get categorical duals for inf-dim Banach spaces is surely related to, though perhaps neither implying nor implied by, the following old result which I think is due to Grothendieck: if X is a Banach space and, for each Banach space E, the usual tensor product of X with E (in the category Vect) has a unique Banach completion, then X is finite-dimensional. More pithily, the only nuclear Banach spaces are the finite-dimensional ones.