Let $\mathbf{A}$ be an Abelian category. Let $X$, $Y$ and $Z$ be objects in $\mathbf{A}$, with $\iota_{1}: X \hookrightarrow Z$ and $\iota_{2}: X \hookrightarrow Y$ monomorphisms. Is it true, and if yes why, that the pushout $P$ of $\iota_{1}$ and $\iota_{2}$ includes into the direct sum $Z \oplus Y$? Because I have been told that this is true since $\iota_{1}$ and $\iota_{2}$ are monomorphisms, but I fail to see it.
By Abelian category I mean a category such that (i) it has the zero object; (ii) it has all binary biproducts; (iii) every morphism has kernel and cokernel; (iv) every monomorphism is a kernel, and every epimorphism is a cokernel.
Edit: I realised that it is not true in general. I would still like to know which other hypotheses I have to add to obtain the result.