Category Theory/(Co-)cones and (co-)limits

Definition (cone):

Let be a category and let be a diagram in . A cone over is an object of , together with morphisms for each , so that for each (so that ) we have

.

Definition (limit):

Let be a category, let be a diagram in and let

Definition (diagonal):

Let be a category, and let be an object of . Further, suppose that the object and the morphisms and constitute a product of with itself in . Then the diagonal (also called "diagonal morphism") is the unique morphism

such that .

Definition (anti-diagonal):

Let be a category, and let be an object of . Further, suppose that the object and the morphisms and constitute a coproduct of with itself in . Then the anti-diagonal (also called "anti-diagonal morphism") is the unique morphism

such that .

Exercises

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    1. 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  .
    2. Prove that any morphism   in   gives rise to a natural transformation  .
    3. 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  .)