I am following Pavel et al book "Tensor Categories". They write that an additive category is a category $\mathcal{C}$ such that:
My problem is with (A3). i know that what they are trying to say is that all bicoproducts exist. But I do not see how that is equivalent to what they wrote: as far as I understand, the coproduct $X_1\rightarrow X_1\sqcup X_2\leftarrow X_2$ is such that for every other pair of arrows $X_1\rightarrow Y\leftarrow X_2$ we have an unique arrow $f$ such that
commutes.
I do not see how to connect this with what they say in (A3).
The author is here using to their advantage the fact that there is an additive structure.
Indeed, let me use the notations of A3, and let $Z$ be an object with maps $f_i: X_i\to Z$. Then consider the map $Y\to Z$ defined by $f=f_1\circ p_1+f_2\circ p_2$.
One may check at once that $f\circ i_1 = f_1\circ p_1\circ i_1 + f_2\circ p_2 \circ i_1 = f_1 + f_2\circ 0 = f_1$ (indeed one may prove that $p_2\circ i_1 = 0$ using the fact that $i_2$ is a monomorphism), and similarly $f\circ i_2 = f_2$, and if $g$ satisfies those equations, then $g= g\circ i_1\circ p_1 + g\circ i_2\circ p_2 = f_1\circ p_1 + f_2\circ p_2 = f$, thus our map $f$ is unique.
This shows that $(Y,i_1,i_2)$ is a coproduct in the usual sense. Similarly you may prove that $(Y,p_1,p_2)$ is a product in the usual sense. This is what makes it a biproduct.