In wikipedia, sheaves were first defined in the case of concrete categories (with usual identity and gluing axioms), then in the general case. (writing it as an "exact" sequence)

Do these two definitions agree? I find the definition for concrete categories case very strange, for if we consider a topological space of two points a,b with discrete topology, and let us consider a sheaf of topological spaces on it that assigns A to a, B to b. According to the "concrete-category-case" definition, we need the global sections to look like $A \times B$ such that the projection maps are continuous and nothing else. But if we look at the "equalizer" definition, we would require the global sections to carry the product topology as well.

So is wikipedia wrong? Or am I misunderstanding something? Thanks!

Edit: There is another not-quite-related question. In wikipedia, for the "equalizer" definition they require the category, where the sheaf's taking values in, to have products. Is this really necessary? In EGA Chapter 0 p.23 for example, the product is just "splitted", and we consider the large family of maps all together. It seems that these two approaches are just the same. Or am I wrong?