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We show how decreasing diagrams introduced in the theory of rewriting systemscan be used to prove coherence type theorems in category theory. We apply thismethod to describe a coherent presentation of the $0$-Hecke monoid$\mathcal{H}(\Sigma_n)$ of the symmetric group $\Sigma_n$, i.e. a presentationby generators, relations, and relations between relations.

Decreasing diagrams and coherent presentations