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In \cite{Schweigert:2019zwt} it was shown how string-net spaces for the Cardybulk algebra in the Drinfeld center $\mathsf{Z}(\mathsf{C})$ of a modulartensor category $\mathsf{C}$ give rise to a consistent set of correlators. Weextend their results to include open-closed world sheets and allow for moregeneral field algebras, which come in the form of$(\mathsf{C}|\mathsf{Z}(\mathsf{C}))$-Cardy algebras. To be more precise, weshow that a set of fundamental string-nets with input data from a$(\mathsf{C}|\mathsf{Z}(\mathsf{C}))$-Cardy algebra gives rise to a solution ofthe sewing constraints formulated in \cite{Kong_2014} and that any set offundamental string-nets solving the sewing constraints determine a$(\mathsf{C}|\mathsf{Z}(\mathsf{C}))$-Cardy algebra up to isomorphism. Hence wegive an alternative proof of the results in \cite{Kong_2014} in terms ofstring-nets.

Cardy Algebras, Sewing Constraints and String-Nets