Uniqueness of morphism (reasoning in categorial language). This question is related to a previous question of mine.
I figured out that maybe the right thing to ask for is for someone to solve one of the problems in Aluffi's book, that way I’ll know what the reasoning should look like.
The following question appears in section 2.3 of Aluffi's book:

Let $\varphi: G \to H$ be a morphism in a category with products.
Explain why there is a unique morphism
$$(\varphi \times \varphi): G \times G \to H \times H.$$
compatible in the evident way with the natural projections.
Next let $\psi: H \to K$ be a morphism in the same category and
consider morphisms between the products $G \times G, H \times H, K \times K$.
Prove: $$(\psi \varphi) \times (\psi \varphi) = (\psi \times \psi)(\varphi \times \varphi).$$

How should a thorough rigorous answer to this question look like? Answers with diagrams are preferred (since this is a major part i still don't get).
Please don't leave any holes in reasoning unless they are totally obvious and if you use diagrams please explain how you reason with them.
I'd like to emphasis that I'm not at all asking this question out of laziness. I'm finding myself a bit lost in this new language and in need of guidance.
Big pluses to anyone who would make me understand the big picture and make it all feel like a bad dream.

The question isn't right.  There isn't a unique morphism $G\times G\to H\times H$, there's a unique morphism... with such and such properties.
I recommend beginning your thinking in the category of sets.  If I have a map of sets $G\to H$, is there a "natural" map $G\times G \to H\times H$ that springs to mind?  Is there some special property that distinguishes this map from any other?
