One of the axioms of additive categories is the following:
(Axiom A3) For any pair of objects $X_1,X_2$ there exist an object $Y$ and morphisms $p_1,p_2,i_1,i_2$
\begin{equation*}X_1\xrightarrow{i_1} Y\xleftarrow{i_2} X_2,\ X_1\xleftarrow{p_1}Y\xrightarrow{p_2}X_2\end{equation*}
with the following properties:
$p_1i_1=id_{X_1}$, $p_2i_2=id_{X_2}$, $p_2i_1=p_1i_2=0$, $i_1p_1+i_2p_2=id_Y$
Gelfand's Homological algebra states that this axiom can be reformulated by saying that two sqaures below are respectively cartesian and cocartesian: $\require{AMScd}$ \begin{CD} Y @>{p_1}>> X_1\\ @Vp_2VV @VVV \\ X_2 @>{}>> 0 \end{CD} and \begin{CD} 0 @>{}>> X_1\\ @VVV @VVi_1V \\ X_2 @>{i_2}>> Y \end{CD} I was able to prove that the condition from the axiom A3 implies that two squares are respectively cartesian and cocartesian but I am unable to prove the opposite direction. Can you please help?