morphisms in $Vect_k^\otimes$

Question

Let $Vect_k^\otimes$ be the tensor category of finite dimensional $k$-vector spaces with the tensor product. What is an example of a morphism $f: V \to W$ in $Vect_k$ which is no morphism in $Vect_k^\otimes$?

Answer 1

The morphisms of ${\rm Vect}_k^\otimes$ are exactly the same morphisms as in ${\rm Vect}_k$.
The tensor category has additional structure, compared to the plain category ${\rm Vect}_k$, namely the tensor functor $$\otimes:{\rm Vect}_k\times{\rm Vect}_k\ \to\ {\rm Vect}_k\,,$$ which is associative up to isomorphism and has a unit.