The path algebra as a left adjoint functor

We consider an intermediate category between the category of finite quivers and a certain category of pseudocompact associative algebras that contains all finite dimensional algebras. We define the completed path algebra and the Gabriel quiver as functors. We give an explicit quotient of the category of algebras on which these functors form an adjoint pair. We show that these functors respect ideals, obtaining in this way an equivalence between related categories.