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

Exercises

 * 1) Let $$\mathcal D$$ be a category such that every two objects of $$\mathcal D$$ have a product. Suppose further that $$\mathcal C$$ is another category, and that $$T: \mathcal C \to \mathcal D$$ is a functor. Let $$Y$$ be an object of $$\mathcal D$$. Use the universal property of the product in order to show that there exists a functor $$S_Y: \mathcal C \to \mathcal D$$ that sends an object $$X$$ of $$\mathcal C$$ to the object $$T(X) \times Y$$ of $$\mathcal D$$.
 * 2) Prove that any morphism $$g: Y \to Z$$ in $$\mathcal D$$ gives rise to a natural transformation $$S_Y \to S_Z$$.
 * 3) Can we weaken the assumption that every two objects of $$\mathcal D$$ have a product? (Hint: Consider the image of the class function on objects associated to the functor $$T$$.)
 * 1) Can we weaken the assumption that every two objects of $$\mathcal D$$ have a product? (Hint: Consider the image of the class function on objects associated to the functor $$T$$.)