Coproducts and products are same in any preadditive category Here is the proof that coproducts and products are same in any preadditive category from the Stack project. $_? Also, I don't see how the mapping they get from $Mor(x,w) \times Mor(y,w)$ to $Mor(z,w)$ is actually a bijection?

1. We have the projections $p:x\times y\to x$ and $q:x\times y\to y$, which satisfy the universal property that any pair $f:a\to x,\ g:a\to y$ correponds to a unique arrow $a\to x\times y$. Now this is applied with $(f,g)=(1,0)$ as for $i:x\to z$ and with $(f,g)=(0,1)$ as for $j:y\to z$ where $0$ is the zero morphism and $1$ is the identity.
By this definition, importantly, we have the following equations: $$p\circ i=1_x,\quad q\circ i=0,\\\ p\circ j=0,\quad q\circ j=1_y$$

2. The mapping $\def\Mor{\mathrm{Mor}}\Mor(x,w)\times\Mor(y,w)\to\Mor(z,w)$ is just given above, call it $\Phi\ :=\ (a,b)\mapsto a\circ p\,+\,b\circ q$.

**Hint:** Use the equations in 1. to find the inverse for $\Phi$.