Continuous Functions on Final Comodels of Free Algebraic Theories

In 2009, Ghani, Hancock and Pattinson gave a tree-like representation of stream processors $A^{\mathbb{N}}\to B^{\mathbb{N}}$. In 2021, Garner showed that this representation can be established in terms of algebraic theory and comodels: the set of infinite streams $A^{\mathbb{N}}$ is the final comodel of the algebraic theory of $A$-valued input ${\mathbb{T}}_A$ and the set of stream processors $Top(A^{\mathbb{N}},B^{\mathbb{N}})$ can be seen as the final ${\mathbb{T}}_A$-${\mathbb{T}}_B$-bimodel. In this paper, we generalize Garner’s results to the case of free algebraic theories.