Is the relative product in the category of Abelian groups the same as the relative product in the category of Groups? I'm not sure if "relative product" is common terminology, but given $N \xrightarrow{f} A \xleftarrow{g} K$ in a category, the relative product is the diagram
$$\begin{array}
A N\times_A K & \stackrel{}{\longrightarrow} & N \\
\downarrow{} & & \downarrow{f} \\
K & \stackrel{g}{\longrightarrow} & A  
\end{array}
$$
that satisfies the universal property of there being a unique morphism from $G \to N\times_A K$ whenever $G$ satisfies the above diagram.
We were told that the relative product in abelian groups was $\{(x,y) \mid f(x) =g(y)\}$, and I'm fairly sure I proved this right, but I noticed that I didn't use commutativity. Does this mean the relative product in group is exactly the same? If I did prove it right, then I believe it is yes, but my gut says it shouldn't be because the coproduct in the two categories is different.
I'm mostly looking for "yes/no" type answers, since if it is false, I want to think about it some more. Also, could someone point me to some literature on this topic? Unless it goes by a different name, I don't believe this is in my text and googling didn't seem to bring up anything.
 A: Yes, this is true (and more generally, the same construction works in pretty much any "concrete category").  I should mention that I have never seen the term "relative product" used to refer to this construction; more common terms are "pullback" or "fiber product".
As a deeper explanation for why relative products look the same (while other constructions like coproducts do not), the inclusion functor from abelian groups to all groups has a left adjoint, namely the functor which takes a group to its abelianization.  Relative products can be described as a type of limit, and right adjoint functors always preserve limits, so the inclusion functor preserves relative products.  On the other hand, coproducts are a kind of colimit, and the inclusion functor does not have a right adjoint, so we cannot expect it to preserve them.
More generally, a whole lot of categories that people like to think about (including both groups and abelian groups) come with a "forgetful" functor $G:\mathcal{C}\to Set$, which has a left adjoint $F:Set\to\mathcal{C}$ taking a set to the object "freely generated" by that set.  In particular, this implies that $G$ preserves all limits, which informally means that limits in $\mathcal{C}$ are constructed "the same way" as in sets.  This can be seen as the reason that in so many familiar categories, products are given by some kind of "cartesian product" construction.  The same reasoning applies to relative products.
