Natural maps arise all the time in algebra and topology and it's important to understand the definition. A canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. A closely related notion is a structure map or structure morphism; the map that comes with the given structure on the object. They are also sometimes called canonical maps.Canonical maps are usually associated with quotient structures, which allow one to generalize the idea of congruence to abstract algebriac objects or topological ones, such as quotient spaces.

The best way to understand what a natural or canonical map is to see some examples.

1) If N is a normal subgroup of a group G, then there is a canonical map from G to the quotient group G/N that sends an element g to the coset that g belongs to. 2) If V is a vector space, then there is a canonical map from V to the second dual space of V that sends a vector v to the linear functional fv defined by fv(λ) = λ(v). 3) If f is a ring homomorphism from a commutative ring R to commutative ring S, then S can be viewed as an algebra over R. The ring homomorphism f is then called the structure map (for the algebra structure). The corresponding map on the prime spectra: Spec(S) →Spec(R) is also called the structure map. 4) If E is a vector bundle over a topological space X, then the projection map from E to X is the structure map.

In this case, the natural map is the indexed ordered pair map of into X x Y . Clearly, the map f that defines C is the indexing map of the ordered pairs in X x Y where and . So that Γ: C → X × Y is defined by Γ() = () where is the unordered indexed pair in C and () an ordered pair in X x Y.

Can you now define the inverse? First,you have to show this map is an isomorphism, which isn't hard.