Category Theory/(Co-)cones and (co-)limits
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- Let be a category such that every two objects of have a product. Suppose further that is another category, and that is a functor. Let be an object of . Use the universal property of the product in order to show that there exists a functor that sends an object of to the object of .
- Prove that any morphism in gives rise to a natural transformation .
- Can we weaken the assumption that every two objects of have a product? (Hint: Consider the image of the class function on objects associated to the functor .)