Biproduct-induced addition of morphisms ($f+g={\nabla }_B⟨f,g⟩$)

In a pointed category with finite product in a category and coproducts in a category and invertible canonical maps $c:A+B\to A\times B$, the unique commutative-monoid enrichment is $f+g=[1_B,1_B]c_{B,B}^{-1}⟨f,g⟩$. Associativity and commutativity follow by comparing fold maps on triple and swapped coproduct injections. Composition distributes by the universal properties. For uniqueness, bilinearity forces $i_1{p}_1+i_2{p}_2=1$ on each biproduct, and composing with the fold and the paired morphism forces the addition formula.