I'm reading Goldblatt's: Topoi. Here:
I am a little confused: He says that we're going to construct the product set without any reference to ordered pairs. This is fine but he says that we are going to construct $A\times B$ and starts talking about it as if it were constructed, when he mentions $p_A: A\times B \to A$. I was expecting it to appear as some kind of last step, what am I thinking wrong?
Goldblatt is not constructing $A\times B$; he is describing what $A\times B$ is from the perspective of category theory, that is, what makes the direct product the direct product. Goldblatt's point is that it doesn't matter how we code ordered pairs (Kuratowski, Hausdorff, Wiener, or otherwise); all that matters is the "global structure," that is, that associated with $A\times B$ are "projection maps" $p_A$ and $p_B$ with the obvious properties. Indeed, he'll go on to show that if $C$ is some other set, with maps $q_A$ and $q_B$ which satisfy the same categorial properties, then $C$ is isomorphic-as-a-set to $A\times B$ - and in fact this isomorphism sends $q_A$ to $p_A$ and $q_B$ to $p_B$.
The point is that in category theory, constructions aren't defined by what they are, but by what they do. Exactly what set $A\times B$ is depends on how I code pairs as sets; that's weird, since the idea of $A\times B$ doesn't make reference to a specific ordered pair setup. The point Goldblatt's making is that underlying the idea of an ordered pair is this picture of a set with a couple arrows coming out of it, satisfying some properties. This is an example of a universal construction.
Replacing a specific set-theoretic construction with a category-theoretic one has some nice intuitive properties; but its real value comes from the impact it has in other categories. We can write down (as Goldblatt does) a categorial description of what the direct product is doing in Sets; but now that transfers to any category! We can speak of a direct product in Groups, or Rings, or (my personal favorite) BanAnaMan, and general arguments around direct products of sets tend to hold in arbitrary categories (note that there's nothing ensuring that direct products exist in arbitrary categories, but the language is always meaningful).