Not every monoidal category is rigid

Let $k$ be a field and let ${\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_k$ be the category of all $k$-vector spaces with the usual tensor product and unit object $k$. This is a monoidal category, and $V:=k[x]$ is infinite-dimensional by [L1].

Assume for contradiction that $V$ had a left dual $(V^{\vee },\mathrm{ev},\mathrm{coev})$. Because $\mathrm{coev}(1)\in V\otimes V^{\vee }$ is a single tensor in an algebraic tensor product, it is a finite sum ${\sum }_{i=1}^m{v}_i\otimes f_i$ with $f_i\in V^{\vee }$. Applying the first zig-zag identity to any $v\in V$ gives $v={\sum }_{i=1}^m\mathrm{ev}(f_i\otimes v)v_i,$ so every vector of $V$ lies in the span of the finite set $\{v_1,\dots ,v_m\}$.

Step 1.2 makes $V$ finite-dimensional, contradicting that $k[x]$ has no finite basis. Therefore $V$ has no left dual, hence is not rigid, so ${\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_k$ is a monoidal category that is not rigid.