Prove that additive functor preserves products and coproducts

Let $\cal A,B$ be additive categories and $F:\cal A\rightarrow B$ be an additive functor. Show that $F$ preserves products and coproducts.

Since product and coproduct of a pair $A,B$ of objects in an additive category are isomorphic, it is enough to prove $F$ preserves coproducts. Let $A,B\in\textrm{Ob}(\cal A),$ let $i_A:A\rightarrow A\oplus B$ and $i_B:B\rightarrow A\oplus B$ be two canonical inclusions. We need to show that $(F(A\oplus B),Fi_A,Fi_B)$ is a coproduct in $\cal B$.

If $\Phi=F\varphi:FA\rightarrow FX$ and $\Psi=F\psi:FB\rightarrow FX$ it is not difficult to show that $F\sigma:F(A\oplus B)\rightarrow FX$ such that $F\varphi=F\sigma Fi_A$ and $F\psi=F\sigma Fi_B$ is unique.

But what if $\Phi:FA\rightarrow FX$ does not lie in image of $F$? Why the needed $\Sigma:F(A\oplus B)\rightarrow X$ exists?

Solution 1:

You have to use the fact that your coproduct $A \oplus B$ is also a product. That is, there are morphisms $p_A$ and $p_B$ from $A \oplus B$ to $A$ and $B$, respectively, such that $(A \oplus B, p_A, p_B)$ is a product. Moreover, the projections interact with the injections by: $$p_A \circ i_A = 1_A, p_B \circ i_B = 1_B, i_A p_A + i_B p_B = 1_{A \oplus B}.$$

Applying your functor to these equations, you can show that $F(A \oplus B)$ remains a product/coproduct.