Let me give you another example: Let X be a set, and Fun(X) = { real valued functions on X }. Then X -> Fun(X) is contra-variant. Indeed, if $F: X -> Y$

Let me give you another example: Let X be a set, and Fun(X) = { real valued functions on X }. Then X -> Fun(X) is contra-variant. Indeed, if $F: X -> Y$ is a map between set, and f \in Fun(Y). Then you have a natural function $f \circ F: X -> IR \in Fun(X)$. This is sometimes called a pull-back of $f$. Functors of the kind Space -> { some stuff* that lives on X } are often contra-variant. E.g. functions, vector bundles, sheaves, differential forms, etc.