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16 1. Categories and Functors Because of this, we will generally write maps F : Z → X ×Y in the form (f, g), where f = pr X ◦ F and g = pr Y ◦ F . One particularly important map is the diagonal map Δ : X −→ X × X defined by Δ = (idX, idX). Exercise 1.35. (a) Show that if one of the products X × (Y × Z) and (X × Y ) × Z exists in C, then so does the other, and they are isomorphic. (b) Let f : A → X and g : B → Y , and suppose that the products A×B and X × Y can be formed in C. Give an explicit definition for the product map f × g : A × B −→ X × Y. Suppose that C is a category with the property that every pair of objects X, Y ∈ C has a product. Then by choosing one product X ×Y for each pair, we see that Exercise 1.35(b) implies that the rules (X, Y ) → X × Y and (f, g) → f × g define a functor of two variables ? × ? : C × C −→ C. Exercise 1.36. Formulate precise definitions of the product C × D of two categories and of a functor of two variables. Some authors use the notation f ×g to denote the map Z → X ×Y with components f and g. But this is wrong: f × g is the image of the ordered pair (f, g) under the functor ? × ? . Problem 1.37. Let X and Y be objects in a category C. (a) Show that, for any map f : X → Y , the diagram X Δ f X × X f×f Y Δ Y × Y commutes. (b) Show that the diagram X × Y Δ id (X × Y ) × (X × Y ) (pr 1 ,pr 2 ) X × Y is commutative.