??maybe filtered cocompleteness comes just from being closed under taking faces, while being closed under intersection is more for the purpose of getting the TAG structure??? ....

??though... seems like you might almost be able to get the TAG structure, too, without needing closure under intersection ... ??? ?? pro-monoidal ... ???? tensor product of categories enriched over abelian group ... here i do sort of seem to be trying to push operad (??...) interpretation of pro-monoidal structure .... ???.... ????? ......

??? trying to think about / visualize translations between dual cones directly in terms of the cone / fan picture ?? ... ... ??? ...