Lang's Algebra, pg 61, fibered products and coproducts of abelian groups I have two distinct questions about this page:
First,

The fibered product of two homomorphisms $f:X\to Z$ and $g:Y\to Z$ is the subgroup of $X\times Y$ consisting of all pairs $(x,y)$ such that $f(x)=g(y)$.

My attempt: First, it is clearly a subgroup of $X\times Y$. So consider $(x,y)\in X\times_Z Y$. We require $(f\circ p_1)(x,y)=f(x)=(g\circ p_2)(x,y)=g(y)$ for the diagram to commute so clearly we must have $f(x)=g(y)$. Is this enough to prove that this fibered product exists?
Second,

The coproduct of two homomorphisms $f:Z\to X$ and $g:Z\to Y$ is the factor group $(X\oplus Y)/W$ where $W$ is the subgroup of $X\oplus Y$ consisting of all elements $(f(z),-g(z))$ with $z\in Z$.

Surely he means to say 'fibered coproduct' or 'pushout' instead of 'coproduct'? I thought the coproduct of abelian groups was a direct sum.
If it does mean pushout, I continue by choosing $z\in Z$ and dutifully following the diagram. To commute, we must have $(q_1\circ f)(z)=(f(z),e_Y)=(q_2\circ g)(z)=(e_X,g(z))$. But how does this translate into needing to quotient out $(f(z),-g(z))$?
I'm very much new to category theory so any help is appreciated.
 A: Second question first: the coproduct of morphisms is the same thing as the pushout. You're right that the coproduct of objects is the direct sum.
Your discussion of the fiber product is quite confused: it's not actually clear to me what you're trying to show. In a literal sense, it does not require proof that the fiber product exists, at least, if you consider that the fiber product is defined as the subgroup of the cartesian product on those pairs $x,y$ mapped to the same point by $f$, respectively, $g$. 
So presumably by "exists" you would prefer to mean "satisfies the universal property," so that it's an actual fiber product categorically. If that's the right interpretation, then you need to show that $X\times_Z Y$ is the universal group mapping to $X$ and $Y$ compatibly with $f$ and $g$, i.e. you need to consider another group $W$ with maps $h,k:W\to X,Y$ and $fh=gk$ and show that $h$ and $k$ factor uniquely through the fiber product.
Your discussion of the pushout is similarly unclear, since I'm not sure what the $q_i$ are-probably addressing the fiber product will be enough at first.
