My question stems from a categorial investigation and precise classification of $\mathtt{FinVect}_\mathbb{k}$, i.e. the category of finite-dimensional vector spaces over a field $\mathbb{k}$, in the wake of an introduction to Frobenius Algebras. I am fairly new to category theory and while I am having an absolute blast, I find the sheer amount of constructs overwhelming at times. Hence my question:
I know that $\mathtt{FinVect}_\mathbb{k}$ is a symmetric monoidal category. Further it is rigid since every object has a left and right dual; since it is symmetric the dual is unambiguous and one calls it a compact closed category (name taken from ncatlab). So far so good. Now looking at Frobenius Algebras there is interesting extra structure like $\mathtt{Alg}_\mathbb{k}$ being its category of monoids and for $A \in \mathtt{Alg}_\mathbb{k}$ one gets $\mathtt{lMod}_A$ (left $A$-modules) as the category of its left actions (or left modules depending on the author). When looking at these, I found that some traditional proofs regarding modules don't generalize to rigid symmetric monoidal categories, especially once they involve the structure of the vector spaces themselves, i.e. that of the objects of $\mathtt{FinVect}_\mathbb{k}$.
Which general concept in category theory grasps the vector space structure and allows for a "descend" of the rigidness of $\mathtt{FinVect}_\mathbb{k}$ onto the elements of $V \in \mathtt{FinVect}_\mathbb{k}$, i.e. for $v \in V$ the dual $v^* \in V^*$?
I have looked at introductory material towards higher category theory and enriched categories but found nothing that seemed directly related although my guts tell me the answer will be either of the two.