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