Non-symmetric PROPs Roughly speaking, a colored PROP is a category in which there are morphisms $x_1,\dotsc,x_n \to y_1,\dotsc,y_m$ with multiple input and output objects. It seems to me that all authors also assume actions of the symmetric groups, so that these should actually be called symmetric PROPs. (This is similar to the convention that topologists mean symmetric operads when they write operads.) Have non-symmetric PROPs been studied in the literature? Or do they have another name? (After all, the last P stands for permutation.) Also, is there a special name for colored PROPS, as there is for colored operads (multicategories)? What about "bimulticategories"? The notion of a polycategory seems to be more restrictive. Or do people usually mean the colored version anyway?

The non-symmetric version is PROs. The colors versions are just called colored. Polycategories are indeed more restrictive.