1.6. Products and Sums 15 Problem 1.32. Verify that the dual of each rule for a category is also a rule for a category, and likewise for functors and natural transformations. Because of Problem 1.32, the dual of a valid proof involving categories, functors and natural transformations is also a valid proof. Thus, the dual of every theorem of pure category theory is automatically also a theorem. Domain- and Target-Type Objects. It often happens that an object of a category is defined in terms of certain category-theoretical properties. These properties usually give special information about the maps out of the object or else they give special information about the morphisms into that object. In the first case, we call the construction a construction of domain type in the second case it is a construction of target type. We will sometimes refer to the results of these constructions as being objects of ‘domain-type’ or of ‘target-type’. The distinction between ‘domain-type’ and ‘target-type’ objects or constructions is important to observe. The dual of a domain-type construction is a target-type construction and vice versa. 1.6. Products and Sums Let X, Y ∈ C. The product of X and Y is an object P together with two morphisms prX : P → X and prY : P → Y (called projections) with the following universal property: if f : Z → X and g : Z → Y are any two morphisms, then there is a unique morphism t : Z → P so that pr X ◦ t = f and prY ◦ t = g. This can be expressed diagrammatically as follows: Z f g ∃! t X P pr X pr Y Y. Since the definition of the product P provides us with a way to understand the maps into P (i.e., it is the target of the hypothetical arrow), products are a target-type construction. There is no guarantee that two given objects in a category C actually have a product or that there will only be one product. Problem 1.33. Suppose X, Y ∈ C and the objects P and Q are both products for X and Y . Show that P ∼ Q. Since any two products are equivalent, we often just choose one of them and denote it by X × Y . Exercise 1.34. Let X, Y ∈ C, and suppose X × Y exists. Explicitly define a bijection morC(Z, X × Y ) ∼ −→ − morC(Z, X) × morC(Z, Y ).

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