What is an intuitive view of adjoints? (version 1: category theory)

I like Wikipedia's motivation for an adjoint functor as a formulaic solution to an optimization problem (though I'm biased, because I helped write it). In short, "adjoint" means most efficient and "functor" means formulaic solution.

Here's a digest version of the discussion to make this more precise:

An adjoint functor is a way of giving the most efficient solution to some optimization problem via a method which is formulaic ... For example, in ring theory, the most efficient way to turn a rng (like a ring with no identity) into a ring is to adjoin an element '1' to the rng, adjoin no unnecessary extra elements (we will need to have r+1 for each r in the ring, clearly), and impose no relations in the newly formed ring that are not forced by axioms. Moreover, this construction is formulaic in the sense that it works in essentially the same way for any rng.

The intuitive description of this construction as "most efficient" means "satisfies a universal property" (in this case an initial property), and that it is intuitively "formulaic" corresponds to it being functorial, making it an "adjoint" "functor".

In this asymmetrc interpretation, the theorem (if you define adjoints via universal morphisms) that adjoint functors occur in pairs has the following intuitive meaning:

"The notion that F is the most efficient solution to the (optimization) problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem which F solves."

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Edit: I like the comment below emphasizing that an adjoint functor is a globally defined solution. If
$G:Cto D$, it may be true that terminal morphisms exist to some $C$'s but not all of them; when they always exist, this guarantees that they extend to define a unique functor $F:Dto C$ such that $F dashv G$. This result could have the intuitive interpretation "globally defined solutions are always formulaic".

Compare this for example to the basic theorem in algebraic geometry that a global section (of the structure sheaf) of $mathrm{Spec} (A)$ is always defined by a single element of $A$; the global sections functor is an adjoint functor representable by the formula $Hom(-,mathrm{Spec}( mathbb{Z}))$, so this is actually directly related.