Cartesian monoidal categories come with projection morphisms that let you go from $A \otimes B$ to $A$ or from $A \otimes B$ to $B.$
I am wondering if there are kinds of monoidal categories which just come with morphisms from $A \otimes B$ to $B$ (so we can only delete on the left).
Are there well studied monoidal categories like this. I had thought about left-rigid monoidal categories where the monoidal unit is the initial object, but they seem too structured.
Any connections with well studied structures would be welcome. I'm also interested in the case where the "delete on the left morphism" can be inverted.