Any function defines a covariant function by postcomposition. Take for example the string-length function (len : string -> int). We can take any other function

Any function defines a covariant function by postcomposition. Take for example the string-length function (len : string -> int). We can take any other function which outputs a string, say (f : X -> string) where X is any fixed type, and produce a new function (g : X -> int) by g(x) = len(f(x)). So the len function defines a functor from the set of functions with signature (X -> string) to the set of functions with signature (X -> int). For concreteness, one can imagine this functor as replacing functions which output strings by functions which output the lengths of those strings instead. However we can also treat (len : string -> int) as a contravariant functor by pre-composing instead of post-composing. Say we have a function (h : int -> Y), then we can form (k : string -> Y) by setting k(s) = k(len(s)). This could be useful if we only cared about whether a strings length were a multiple of 5, say. Some of the above are lies: in order to call these functors in the mathematical sense (or the Haskell sense) you need to phrase things in just the right way. But I think it gets the idea across about how simple the difference between co- and contra- variance can be, with the example of replacing f(x) by g(f(x)) or by f(g(x)).