Let TOP(2) be the category whose objects $(X,A)$ are pairs of topological spaces and whose morphisms $f:(X,A) \to (Y,B)$ are continuous maps $f:X\to Y$ such that $f(A) \subset B$. If I am not mistaken,the products in this category are given by $(X,A) \times (Y,B) \cong (X\times Y,A \times B)$. Am I right about it ?
I am facing another related problem. In the book 'Algebraic Topology' by Tammo Tom Dieck, the author mentions on p. 36 that
The assignment $H: (x,t) \to (\alpha(x,t)^{-1}(1+t) x, 2-\alpha(x,t)) $ where $\alpha(x,t)=max(2||x||,2-t)$ yields a homeomosphism of pairs $(D^n,S^{n-1}) \times (I,0) \cong D^n \times (I,0)$.
I am able to verify that the map is continuous, injective and surjective(with the understanding that the range is $D^n \times I$) but I do not understand what is meant by $D^n \times (I,0)$. The first factor here is a member of TOP while the second factor is a member of TOP(2). How could one possibly talk about the product of objects belonging to two different categories ?
Tom Dieck is using the notation $(X,A)\times (Y,B)$ to mean $$\left(X\times Y, (X\times B)\cup(A\times Y)\right)$$ (see page 32), although as he says there it's not the categorical product (which your interpretation is).
I think he's also identifying a space $X$ with the pair $(X,\emptyset)$, although I can't see where he says so explicitly. At least, the Proposition you mention and the diagram illustrating it make sense if you take $D^n\times (I,0)$ to mean $(D^n,\emptyset)\times (I,0)$, which is just $(D^n\times I, D^n\times 0)$ according to his meaning for the product of pairs.