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It is shown how the graph category of Borisov and Manin can be constructedfrom (a variant of) the graph category of Joyal and Kock, essentially byreversing the generic morphisms. More precisely, the morphisms in theBorisov-Manin category are exhibited as cospans of reduced covers andrefinement morphisms.

Cospan construction of the graph category of Borisov and Manin