TITLE: What exactly is a natural map
QUESTION [5 upvotes]: I have numerous questions on my abstract homework asking me to define "the natural map", though i don't see reference to it in my textbook. 
Let X and Y be sets and let C be the set {f : {1,2}→ X ∪Y|f(1) ∈ X and f(2) ∈ Y} (1) Deﬁne the natural map Γ: C → X × Y (2) Deﬁne the map $Γ^{−1}$: X ×Y → C . 
I don't understand what is being asked of me. Also:
For a set S deﬁne the natural isomorphism Char: P(S) →F(S,{0,1}). For A ⊆ S denote the function Char(A) by χA. (2) Deﬁne Char−1: F(S,0,1) →P(S) 
I know that an isomorphism is a function that perserves structure between two structurally-the-same algebraic thingies. 
Can someone give me an intuitive definition of what all this natural stuff is about?

REPLY [6 votes]: A natural map is a map which you are too lazy to define.

REPLY [5 votes]: Intuitively speaking, to say that a map is "natural" usually just means it's defined independently of any choices, so that if you and I are given the same data, we will come up with the same map. 
More rigorously, most things that are called "natural maps" can be formulated in the language of category theory as natural transformations between functors.