TITLE: Coproducts and products are same in any preadditive category
QUESTION [5 upvotes]: Here is the proof that coproducts and products are same in any preadditive category from the Stack project.

I have few questions regarding the above proof.
I don't understand what do they mean by morphism corresponding to $(0,1)$?
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?

REPLY [4 votes]: 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:
$$p\circ i=1_x,\quad q\circ i=0,\\ p\circ j=0,\quad q\circ j=1_y$$
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 $\Phi$.