I was looking carefully at all the definitions, trying to understand exactly what was going on in [this question on categorical duals in Banach spaces](https://mathoverflow.net/questions/10789/categorical-duals-in-banach-spaces). It seems that in the category of Banach spaces with contractions, only finite dimensional Banach spaces have duals **in the sense of category theory**. This is because one needs morphisms

$$
I \to A \otimes A^\*
$$

and

$$
A \otimes A^\* \to I
$$

where the first is like the inclusion of the identity and the second like the trace. This means that the identity map on $A$ has to have a trace, and that only works for finite dimensional Banach spaces.

So, my question: am I right? Or have I misunderstood something here?