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We give further insights into the weighted Hurwitz product and the weightedtensor product of Joyal species. Our first group of results relate the Hurwitzproduct to the pointwise product, including the interaction with Rota--Baxteroperators. Our second group of results explain the first in terms ofconvolution with suitable bialgebras, and show that these bialgebras are infact obtained in a particularly straightforward way by freely generating frompointed coalgebras. Our third group of results extend this from linear algebrato two-dimensional linear algebra, deriving the existence of weighted Hurwitzmonoidal structures on the category of species using convolution with freelygenerated bimonoidales. Our final group of results relate Hurwitz monoidalstructures with equivalences of of Dold--Kan type.

Coalgebras governing both weighted Hurwitz products and their pointwise transforms