Here's a description which is close to the way this arose in math. It's pretty simple. Maybe it will be helpful for some people to see it removed from programmi

Here's a description which is close to the way this arose in math. It's pretty simple. Maybe it will be helpful for some people to see it removed from programming issues. (Maybe not.) From concreteness, let's suppose we're talking about sets and functions between sets. (The same thing works in an arbitrary category.) Thus, if X and Y are sets, Hom(X, Y) denotes the set of functions (morphisms) from X to Y. Suppose you have a fixed function f: A -> B. You can compose it with functions into A or out of B. If g: C -> A maps into A, then f o g (using "o" to denote composition) gives a function from C to B: that is, f o g: C -> A -> B. We started with a function from C to A and wound up with a function from C to B. So we actually have a function (functor) Hom(C, A) -> Hom(C, B), which is often denoted Hom(C, f). We say that Hom(C, -) is covariant, because it respects the direction of arrows in the second slot. (f went from A to B, and Hom(C, f) goes from Hom(C, A) to Hom(C, B).) If h: B -> D maps out of B, then h o f gives a function from A to D: that is, h o f: A -> B -> D. So we actually have a function from Hom(B, D) -> Hom(A, D) which is often denoted Hom(f, D). We say that Hom(-, D) is contravarient, because it reverses the direction of arrows in the first slot. (f went from A to B, but Hom(f, D) goes from Hom(B, D) to Hom(A, D).) g f h C ---> A ---> B ---> D Thus, Hom(-, -) is actually a bifunctor which is covariant in one variable and contravariant in the other. Contravariant functors can be regarded as covariant functors on the opposite category. What is happening with Hom is a prototype for many of the ways that "covariance" and "contravariance" occur in math; for example, covariant and contravariant tensors. (The vector dual space function Hom(-, K) [where K is the ground field] is contravariant.) One of the earliest descriptions of category theory (including variance) is in: Samuel Eilenberg and Saunders MacLane, "General Theory of Natural Equivalences". Transactions of the American Mathematical Society, Vol. 58, No. 2, (Sep., 1945), pp. 231-294 It's actually fairly readable. There have been many books and articles on category theory since then, and many specifically directed toward computer science (e.g. Michael Barr and Charles Wells, "Category Theory for Computer Science" - https://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf https://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf).