Coproduct of $(0,1)$-Algebras I am trying to find the coproduct of $(\mathbb {Z},0,+1) $ with itself in the category of $(0,1) $-Algebras. Finding $\mathbb {N}\sqcup\mathbb {N} $ was easy, since $\mathbb{N} $ is initial. But I don't know how this coproduct looks in general.

Coproducts can be computed by means of generators and relations. In this case, it is not hard to see that the coproduct in question is two copies of $\mathbb{Z}$ glued along the non-negative integers, i.e. the algebra whose underlying set is $(\mathbb{Z} \times \{ 0, 1 \}) / \sim$ where $(n, m) \sim (n', m')$ if and only if $n = n'$ and either $m = m'$ or $n \ge 0$, with the distinguished constant being $(0, 0)$ and the operation being $(n, m) \mapsto (n + 1, m)$.
