To follow up on Dan's answer, a geometric construction of the Saneblidze--Umble diagonal is given by Masuda--Tonks--Thomas--Vallette in http://www.numdam.org/item/JEP_2021__8__121_0/, and an alternative description of the SU combinatorics is given here https://arxiv.org/abs/2308.12119 (see in particular Theorem 5.17).
A similar construction gives a tensor product of A-infinity morphisms https://jep.centre-mersenne.org/articles/10.5802/jep.221/. In fact, there are two distinct formulas for the tensor product of A-infinity morphisms (see Section III in https://arxiv.org/abs/2308.12119), whereas there seems to be only one for the product of A-infinity algebras...!