# Show that this duality functor is not an isomorphism between categories

I am stuck in this exercise.

"Let be $$k$$ a division ring, and $$\textbf{vect-k}$$ (resp. $$\textbf{k-vect})$$ te category of all right (left) vector spaces over $$k$$ that are of finite dimension, and let D:=$$Hom(-, \hspace{0.1cm} _{k}k_{k}):\textbf{vect-k}\rightarrow \textbf{k-vect}$$ the usual duality functor (i.e. the controvariant functor that sends every $$V_{k}$$ in its dual $$_{k}V^{*}$$ and every linear mapping $$f:V_{k}\rightarrow W_{k}$$ in its transposed $$f^{*}:\hspace{0.1cm }_{k}W^{*}\rightarrow \hspace{0.1cm} _{k}V^{*}$$). Show that D is not an isomorphism of categories between $$\textbf{vect-k}$$ and $$(\textbf{k-vect})^{op}$$."

• What are your thoughts on the problem? What have you tried? Nov 9, 2019 at 14:39
• I was able to show that D is certainly a duality, i.e. a full, faithful and essentially surjective functor, but I've no idea to get out of this question. An isomorphism of categories means that I can find another functor G such that GF and FG are the identities functors (in their respectively domains), so I think I could suppose that such G exist and then find some contraddiction, but I don't see the contraddiction... Nov 9, 2019 at 14:47

The obstruction lies in cardinality. Prove that: either $$V^*$$ is finite-dimensional or of uncountable dimension. Therefore, $$(\bullet)^*$$ cannot be essentially surjective.