I'm trying to understand the use of adjoint functors and came across the interpretation of them as the optimal solution to some problem. But am trying to wrap my head around how to connect them with non-categorical concepts of optimisation.
From my current understanding this means if I have an adjoint pair $F \dashv G$ then the process prescribed by the associated universal morphisms from $Y$ to $G$ is the most efficient solution to the problem posed by $G$.
My question is how to take some functor $G$ and work out the meaning of the associated problem. Following the example used on the wikipedia page where $G$ is the forgetful functor from the category ring to rng. The associated optimisation problem seems to be, what is the fewest extra elements and relations I need to add to a rng so that I can label one of the elements as an identity and have it obey identity relations. However I feel like this phrasing of the problem isn't obvious when you just say $G$ is the forgetful functor.
Is there some process or heuristic so that I can take a functor $G$ and describe it as a optimisation problem in a less categoric way? Furthermore what kind of functors $G$ correspond to problems such as minimise this function?