Let $Vect_f$ be the category of finite-dimensional vector spaces. This category comes with a very well-behaved duality functor. Now the ind-completion of this category (if I understand correctly) is the category $Vect$ of all vector spaces, and the pro-completion will then be the dual categor, $ProVect_f$ (which includes things like formal power series).

Suppose I take an object $V$ in $ProVect_f$. Then $V\otimes V^*$ can be viewed as either an object of the ind-completion of $ProVect_f$ or an object of the pro-completion of $Vect = IndVect_f$. My question is whether it is meaningful to view $V\otimes V^*$ as an algebra object in an appropriate monoidal category.