I have previously spoken about how a mild generalization of the Batanin-Markl notion of operadic category is equivalent to a certain type of skew monoidal structure on a power of Set, and that the actual notion of Batanin-Markl corresponds to the case where the skew monoidal category is left normal (the left unit maps are invertible). I have also previously spoken about how, under mild conditions, one can replace a skew monoidal category by one which has the same category of monoids, but is right normal. If the original skew monoidal category arises from a (generalized) operadic category, the "right normalization" may not.
In this talk, I describe how, under mild conditions, one can replace a skew monoidal category by one which is left normal. This time the category of monoids is not the same (although the category of comonoids is). But this time the original skew monoidal category corresponds to an operadic category, then so does the "left normalization". This allows one to replace a generalized operadic category by a genuine (that is, a Batanin-Markl) one.
I also give some examples related to the Borisov-Manin category of graphs.