= Solution
A <semi-additive category> is a category whose hom-sets are <commutative monoid>[commutative monoids] and whose composition is additive in each variable, with finite products and coproducts.
Suppose first that $P=A\times B$ is a binary <product in a category>, with projections $p_1,p_2$. The zero morphisms and the product property define maps
$$
i_1:A\to P,qquad i_2:B\to P
$$
by
$$
p_1i_1=1_A,quad p_2i_1=0,qquad
p_1i_2=0,quad p_2i_2=1_B.
$$
The two projections of $i_1p_1+i_2p_2$ equal those of $1_P$, so
$$
i_1p_1+i_2p_2=1_P.
$$
For $f:A\to C$ and $g:B\to C$, the map
$$
h=fp_1+gp_2:P\to C
$$
satisfies $hi_1=f$ and $hi_2=g$. If $k$ has the same restrictions, then
$$
k=k(i_1p_1+i_2p_2)=fp_1+gp_2=h.
$$
Thus $(P,i_1,i_2)$ is also the binary coproduct. The dual argument starts from a coproduct and makes it a product. Hence binary products and coproducts coincide canonically as <biproduct>[biproducts].
Let $f,g:A\rightrightarrows B$ be a <reflexive pair> in an <additive category>, with $fr=gr=1_B$. For every object $C$, regard $x:C\to A$ as an arrow from $fx$ to $gx$ between objects of $\mathcal C(C,B)$. The identity at $b:C\to B$ is $rb$.
If $gx=fy$, define the composite by
$$
x\circledast y=x+y-rgx.
$$
Its source and target are
$$
f(x\circledast y)=fx,qquad g(x\circledast y)=gy.
$$
The identities follow from
$$
rf x\circledast x=x,qquad x\circledast rgx=x,
$$
and associativity follows immediately by expanding both iterated composites and using the matching equations. The inverse of $x$ is
$$
x^{-1}=rfx+rgx-x,
$$
whose source is $gx$, whose target is $fx$, and whose two composites with $x$ are the appropriate identity arrows. These formulas are natural in $C$, so the <Yoneda lemma> identifies them with structure morphisms in $\mathcal C$. The pair is therefore an internal groupoid, proving that every <reflexive pair in an additive category is an internal groupoid>.
This fails for semi-additive categories. In the category of <commutative monoid>[commutative monoids], let
$$
R=\{(m,n)\in\mathbb N^2:m\leq n\}
$$
under coordinatewise addition. The two projections $f,g:R\rightrightarrows\mathbb N$ have the common splitting $r(n)=(n,n)$, so they form a reflexive pair. Its underlying reflexive graph is the usual order category on $\mathbb N$: there is an arrow $m\to n$ exactly when $m\leq n$. If it were an internal groupoid, the arrow $0\to1$ would have an inverse $1\to0$, but $(1,0)\notin R$. Therefore this reflexive pair is not an internal groupoid, and “additive” cannot be weakened to “semi-additive.”
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