A category with finite biproducts is enriched in commutative monoids

Let ${\mathrm{\nabla }}_B:B\oplus B\to B$ be the copairing $[1_B,1_B]$, and for $f,g:A\to B$ define $f+g:={\mathrm{\nabla }}_B⟨f,g⟩$. Let $0_{A,B}$ be the zero morphism through the zero object supplied by [L2]. These constructions are available because $B\oplus B$ is simultaneously a product and a coproduct by [L1].

Commutativity follows from the symmetry isomorphism $\tau :B\oplus B\to B\oplus B$ in [L3]: one has $\tau ⟨f,g⟩=⟨g,f⟩$ and ${\mathrm{\nabla }}_B\tau ={\mathrm{\nabla }}_B$, so $f+g={\mathrm{\nabla }}_B⟨f,g⟩={\mathrm{\nabla }}_B⟨g,f⟩=g+f$. The unit law follows from the canonical identifications $B\oplus 0\cong B\cong 0\oplus B$ in [L3], which turn $⟨f,0_{A,B}⟩$ and $⟨0_{A,B},f⟩$ into the evident copies of $f$.

Under the canonical associativity isomorphism of [L3], both $(f+g)+h$ and $f+(g+h)$ are the composite of the triple pairing $⟨f,g,h⟩:A\to B\oplus B\oplus B$ with the triple codiagonal $[1_B,1_B,1_B]:B\oplus B\oplus B\to B$. Hence the addition is associative.

If $k:B\to C$, then $k(f+g)=k{\mathrm{\nabla }}_B⟨f,g⟩={\mathrm{\nabla }}_C⟨kf,kg⟩=kf+kg$, because postcomposing a coproduct copairing with $k$ gives the copairing of the postcomposites. Dually, $(f+g)h=fh+gh$ for every $h:X\to A$, because precomposing a product pairing with $h$ gives the pairing of the precomposites. Thus composition is bilinear.

Steps 2.1, 2.2, and 2.3 show that each hom-set is a commutative monoid with bilinear composition. So the category is enriched in commutative monoids.