= Biproducts of complete join-semilattices
In the <category of complete join-semilattices>, hom-sets have pointwise join as addition and the constant-bottom map as zero, making it a <semi-additive category>. For any family $A_i$, the cartesian product with coordinatewise joins is also its coproduct. The injections put an element in one coordinate and bottom in all others. A family of maps $f_i:A_i\to B$ extends uniquely by $(a_i)\mapsto\bigvee_i f_i(a_i)$, since each tuple is the join of its coordinate injections. Thus all set-indexed canonical product-coproduct comparisons are invertible. The two-element chain has distinct identity and zero morphisms, so this category is not trivial and not additive.
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