Given the following diagram, I want to show that if $f$ is injective, then $\alpha$ is injective, and then if $f$ is surjective, $\alpha$ is surjective.
$\begin{array} XX & \stackrel{\alpha}{\longrightarrow} & C \\ \downarrow{\beta} & & \downarrow{g} \\ B & \stackrel{f}{\longrightarrow} & A \end{array}$
Before I begin to prove either of these statements, I want to make sure I am defining each module and map correctly. I believe I want $X=(B\oplus C)/S$, where $S=\langle f(b),-g(c)\rangle$, and $\alpha$ and $\beta$ to be projection maps, where $\alpha((b,c)+S)=c$, and $\beta((b,c)+S)=b$.
For my proof, here is my idea:
I ultimately want to prove that if $\alpha((b,c)+S)=0$, this implies $c=0$. So I choose some $(b_1,c_1)+S\in X$ such that $\alpha((b_1,c_1)+S)=0$. Then, $(b_1,c_1)\in\ker(\alpha)$. At this point, I think I want to be able to say something about $(b_1,c_1)$ using the equivalence relation $f(b)=g(c)$, but I am not sure how to go about this. Any suggestions would be helpful!