Functor in $\mathbf{Ban}$ that puts exact sequences into exact sequences

Question

Let $F$ - is functor in category of Banach spaces $\mathbf{Ban}$ with follow property: $f_n : A_n \to A_{n+1}$ exact sequence iff $F f_n : F A_n \to F A_{n+1}$ is exact sequence. Is it true, that $F$ naturally equivalent to functor of Banach adjoint $\mathbf{(\cdot)^*}$ or identity functor $\mathbf{1}$?

Answer 1

This is definitely not true. For instance, you could take $F(A)=A\oplus A$ (with any reasonable choice of norm). More generally, you should morally expect any functor of the form $F(A)=B\otimes A$ for some fixed nonzero $B$ to have this property, though I don't know enough about tensor products of infinite-dimensional Banach spaces to be able to say how much sense this makes if $B$ is infinite-dimensional.

(Also, the adjoint functor is not even a functor $\mathbf{Ban}\to\mathbf{Ban}$; it is a functor $\mathbf{Ban}\to\mathbf{Ban}^{op}$, i.e. a contravariant functor from $\mathbf{Ban}$ to itself.)