How Should I Prove this Function is Well-defined?

Question

I am trying to prove that the dual sum of P$_1$ and P$_2$ is projective iff P$_1$ and P$_2$ are projective. I am done with every aspect of the proof. I have this diagram which commutes:

Now, let h: P$_1$$\to$C. We define the g in the diagram as g((p$_1$,p$_2$)) = h(p$_1$).

My question is, how do I prove that this is well-defined?

Thanks in advance.

Note: I just noticed I did not put f in the diagram. We have the surjective homomorphism f:B$\to$C.

Note: We are supposed to have C$\to$O in the diagram as well.