Finite Coproduct in Preadditive Category is Biproduct

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Binary coproducts

Let $A$ be a preadditive category.

Let $a_1, a_2$ be objects of $A$.

Let $(a_1 \sqcup a_2, i_1, i_2)$ be their binary coproduct, assuming it exists.

Let $p_1 : a_1 \sqcup a_2 \to a_1$ be the unique morphism with:

$p_1 \circ i_1 = 1 : a_1 \to a_1$
$p_1 \circ i_2 = 0 : a_1 \to a_2$

Let $p_2 : a_1 \sqcup a_2 \to a_2$ be the unique morphism with:

$p_2 \circ i_1 = 0 : a_2 \to a_1$
$p_2 \circ i_2 = 1 : a_2 \to a_2$

where $1$ is the identity morphism and $0$ is the zero morphism.

Then $(a_1 \sqcup a_2, i_1, i_2, p_1, p_2)$ is the binary biproduct of $a_1$ and $a_2$.

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