The context for this question is this paper.
This is a sanity check for me since I'm still getting used to the language of category theory. The unit of the category $\operatorname{Vect(k)}$ is the field $k$ since if we let $V\in \operatorname{Vect(k)}$ then $V\otimes k\cong V$. I.e., for all $v\in V$ and $c\in k$ we have $v\otimes c\in V$, right?
Yup. In particular, the basic tensor $v\otimes c$ is equal to the basic tensor $(c\cdot v)\otimes 1_k$. From this it follows that the map $v\mapsto v\otimes 1_k$ is an isomorphism (it's obviously linear and injective, the only possible issue is surjectivity).