I am trying to come up with an isomorphism of $X$ and ker$(f \oplus g)$, where $f \oplus g: X \oplus Y \to Z$, and $f: X \to Z$ is a homomorphism and $g:Y \to Z$ is an isomorphism of commutative groups.
Somehow I am totally out of ideas. Would be grateful for any kind help.
Consider the map $\pi : \ker(f\oplus g)\to X$ given by $(x,y)\mapsto x.$
We have $\ker\pi = \{(x,y)\in X\oplus Y\mid x = 0\textrm{ and }f(x) + g(y) = 0\}$ But, if $x = 0,$ then we must have $g(y) = -f(0) = 0,$ and as $g : Y\to Z$ is an isomorphism, there is a unique such $y\in Y;$ namely, $y = 0.$ This shows that $\pi$ is injective.
On the other hand, let $x_0\in X.$ Then as $g : Y\to Z$ is an isomorphism, there exists a unique $y_0\in Y$ such that $g(y_0) = -f(x_0).$ This means that $f\oplus g(x_0,y_0) = f(x_0) + (-f(x_0)) = 0,$ so $(x_0,y_0)\in\ker(f\oplus g).$ As $\pi(x_0,y_0) = x_0,$ $\pi$ is also surjective.