I am trying to understand the article "Complete Axioms for Categorical Fixed-point Operators" where they introduce what an iteration operator is. This is basically, a family of functions $$ (\cdot)^\dagger : \mathcal{D}(X \times A, A) \to \mathcal{D}(X, A)$$
They start imposing a series of axioms which they say come from S. Bloom and Z. Esik's Iteration Theories. However, in their references, I only find iteration operators being defined in terms of coproducts: $$ (\cdot)^\dagger : \mathcal{D}(n, n + p) \to \mathcal{D}(n, p) $$
I am confused by this, how is that they change from product to coproduct? And domain to target? Is it the same to have this iteration operators in the sense of Simpson and Plotkin as in the sense of Bloom and Esik?