category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
Given a field $k$, the category $Vect_k$ is category whose objects are vector spaces and whose morphisms are linear map.
If the field $k$ is understood, one often just writes $Vect$.
The study of $Vect$ is called linear algebra.
The full subcategory of Vect consisting of finite-dimensional vector spaces is denoted $\Fin Vect$.
This is a compact closed category (see here).
$\Fin Vect$ is where most of ordinary linear algebra lives, although much of it makes sense in all of $Vect$. See also at finite quantum mechanics in terms of dagger-compact categories.
On the other hand, anything involving transposes or inner products really takes place in $Fin$ Hilb.