Generally, from what I've seen, for all of the practical programming examples, an `F[A]` (for F covariant) can be considered to be a type that produces zero-or-more `A`s. Thought of properly, F maps types of values to some type of producer of values, and morphisms to postcomposition of the underlying function. F contravariant maps types to some type of consumer for that type and morphisms to precomposition.

I've seen people make analogies for covariant functors as containers (typical programming jargon using "functor" to mean the values of the types F[A]), but that's not quite right because of things like Future and IO. But containers are producers (they can produce the values they contain), so that characterization seems to work.

It's possible that what I mean can be turned into a statement like "all functors used in programming are representable", but I'm not sure.