# Showing the intrinsic addition of an Abelian Category is associative

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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This is an old question, but it seems very hard to find an actual proof in literature (I couldn't find any). If you could show bilinearity of the composition and that $0$ is the neutral for $+$, you can easily show $\langle f+g,h \rangle = \langle f,0\rangle + \langle g,h\rangle$ and $[\mathrm{id},\mathrm{id}] = \delta_1+\delta_2$, and from there you have $$(f+g)+h = (\delta_1 + \delta_2)\circ \langle f + g, h \rangle = (\delta_1 + \delta_2)\circ(\langle f,0\rangle + \langle g,h\rangle) = (f+0)+(g+h).$$