Now consider a related construction of the double dual $V^{} = \mathrm{Hom}(\mathrm{Hom}(V,\mathbbe{k}),\mathbbe{k})$ of $V$. If $V$ is finite dimensional, then the isomorphism $V \cong V^$ is carried by the dual vector space functor $(-)^ \colon \textup{\textsf{cat}}\mathbbe{k}^\mathrm{op} \to \textup{\textsf{cat}}\mathbbe{k}$ to an isomorphism $V^* \cong V^{}$. The composite isomorphism $V \cong V^{}$ sends the basis $e_1,\ldots, e_n$ to the dual dual basis $e_1^{}, \ldots, e_n^{**}$.