I am not sure, but I think that the answer is that you can’t say anything interesting with just ◃, but can maybe say interesting things with ◃+ and ◃×, which I am about to introduce. In the post that just went up, ◃+ is the relationship between one of the components and a sub-sum, and ◃× is the relationship between one of the components and a sub-tensor.◃ is the transitive closure of ◃+ and ◃×.
I think that if C◃+D, then there is a nice morphism from C to D, and if C◃×D, there is a set of nice morphisms from C to D, but in some degenerate cases that set is empty, which is how I constructed a counter example in my other comment.
I am not sure, but I think that the answer is that you can’t say anything interesting with just ◃, but can maybe say interesting things with ◃+ and ◃×, which I am about to introduce. In the post that just went up, ◃+ is the relationship between one of the components and a sub-sum, and ◃× is the relationship between one of the components and a sub-tensor.◃ is the transitive closure of ◃+ and ◃×.
I think that if C◃+D, then there is a nice morphism from C to D, and if C◃×D, there is a set of nice morphisms from C to D, but in some degenerate cases that set is empty, which is how I constructed a counter example in my other comment.