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

Exercises

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