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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.

In the category of complete join-semilattices, define addition of arrows pointwise byand let the zero arrow be the constant-bottom map. These form commutative monoids. Pointwise join of join-preserving maps again preserves all joins, including the empty join, and composition is additive in each variable. Thus the category has the required semi-additive enrichment.

For an arbitrary set-indexed family $(A_i)$, the product poset $P={\prod }_i{A}_i$, with coordinatewise joins, is the product in a category. Define ${\nu }_i:A_i\to P$ by putting an element in coordinate $i$ and bottom in all others. Given join-preserving $f_i:A_i\to B$, defineThis preserves arbitrary joins by interchanging the two joins, and $h{\nu }_i=f_i$. It is unique because every tuple is ${\bigvee }_i{\nu }_i(a_i)$. Therefore $P$ is also the coproduct in a category with these injections, and its canonical coproduct-to-product map is the identity. This proves the biproducts of complete join-semilattices property, even for arbitrary set-indexed families, and in particular makes this a semi-additive category with the hypotheses of part (ii).

The two-element chain has distinct identity and constant-bottom endomorphisms, so the category is not equivalent to $1$. Its addition is idempotent rather than cancellative; for example $1_A+1_A=1_A$ itself supplies an absorbing endomorphism. Hence