# Showing the intrinsic addition of an Abelian Category is asociative

Let $\mathcal A$ be an abelian category. Let $(P\oplus Q,\eta_1,\eta_2)$ be the coproduct of $P$ and $Q$, then if $\delta_1=[id_P,0]$ and $\delta_2=[0,id_Q]$, then $(P\oplus Q,\delta_1,\delta_2)$ is the product of $P$ and $Q$.

Now let $f,g:X\rightarrow Y$ be arrows in the category, then consider a coproduct $\iota_i:Y\rightarrow Y\oplus Y$, $i=1,2$, of $Y$ with itself, let $\delta_i:Y\oplus Y\rightarrow Y$, $i=1,2$, be the associated product with respect to the $\iota_i'$s. Let $\sigma =[id_Y,id_Y]$, then define $f+g=\sigma \langle f,g\rangle$.

Now I have to show $+$ is associative, my teacher hinted that I only needed to show that $h(g_1+g_2)f=hg_1f+h_1g_2f$, and apply this equality suitably to $\delta_1,\delta_2,\delta_1+\delta_2$, but i'm nowhere with this, and I've tried a lot, so i'd like to get more light on this exercise. Thank you.

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