Is the coevaluation map of finite-dimensional vector spaces always a mono?

Question

Let $V$ be a finite-dimensional vector space over a field $k$ and denote by $V^*$ its dual.

Write $\eta:k\to V\otimes V^*$ for the "coevaluation map", i.e the unit of the adjunction making $V$ a dualizable object. Is $\eta$ always a monomorphism?

Answer 1

If $V$ is zero-dimensional, then the answer is no.

If $V$ is non-zero, then the answer is yes. If $\{v_1, \ldots, v_n\}$ is a basis of $V$, then the coevaluation map is defined by $$ k\to V\otimes V^*: \lambda \mapsto \sum_{i=1}^n \lambda v_i\otimes v_i^*,$$ so it is non zero, and thus injective (since its domain is one-dimensional).