Q:

Fibered coproducts in $\mathsf{Ab}$

Aluffi (II.3.9) asks

Show that fiber products and coproducts exist in $\mathsf{Ab}$. (Cf. [ exercise on fiber products and coproducts in $\mathsf{Set}$]). 

The equalizers for $\mathsf{Ab}$ and $\mathsf{Grp}$ seem to be the same as for $\mathsf{Set}$, which gives the fiber products immediately. 
I couldn't figure out the coequalizers for $\mathsf{Ab}$, so I "cheated" by looking them up on Wikipedia.  They turn out to be quotient groups of a certain sort that seem to make sense (I haven't fully worked out the details yet, but one key point seems to be that in $\mathsf{Ab}$, as opposed to $\mathsf{Grp}$, the difference of two homomorphisms is always a homomorphism).  But at this point in the text, Aluffi hasn't yet introduced quotient groups!  So I'm wondering if there's some way to construct the fiber coproduct without involving the coequalizer.  I'm looking more for a hint as to what direction to take than for a full answer. 

A:

You have to understand quotients before dealing with fibered coproducts (aka pushouts)!  After all, the whole idea of a pushout (or colimit in general) is to identify things, and this works via quotients.  I doubt that there is another explicit construction (OK using Freyd's representability criterion you get all colimits of algebraic structures for free, but this is not explicit). 
As a general rule: Don't study any general theory (such as category theory) before having studied enough examples and basic constructions.