Consider we are in an abelian category $\mathcal{C}$ then I define an object "M" in $\mathcal{C}$ as finitely presentated if there is an exact sequence $P_{1} \to P_{1} \to M \to 0$ such $P_{0}$ and $P_{1}$ are projective finetely generated objects in $\mathcal{C}$, where by and finitely generated object $P_{0}$ for example I mean that for every epimorhism
$$\oplus_{i \in I}Hom_{C}(-,C_{i}) \to P_{0} \to 0 $$ there is always a finite set $J \subset I$ such
$$\oplus_{i \in J}Hom_{C}(-,C_{i}) \to P_{0} \to 0 $$ is also an epimorhism.
So I want to prove thath if $M_{1}$ and $M_{2}$ are finitely presentated objects then $M_{1} \oplus M_{2}$ is finitely presentated where $M_{1} \oplus M_{2}$ is the coproduct of $M_{1}$ and $M_{2}$ by thath i mean I have a family of morphism $\lbrace i_{i}:M_{i} \to M_{1} \oplus M_{2} \rbrace_{i=1,2}$ which satisfy the universal property of the coproduct .
So Im really stucked trying to construct an exact sequence $P_{1}'' \to P_{0}'' \to M_{1} \oplus M_{2}$ such $P_{1}'$ and $P_{0}''$ are projective finitely generated objects, can anyone help me out by sketching me a glimpse of the proof or just guide me a little bit?
Thanks