A 2-polygraph is regular if its 2-cells have non-degenerate, “interval-shaped” input and output boundaries. A merge-bicategory is a regular 2-polygraph with an algebraic composition of 2-cells, where the composable diagrams are the “disk-shaped” ones. A merge-bicategory is representable if it contains enough “divisible” 1-cells and 2-cells, satisfying certain universal properties.

I will show that representable merge-bicategories and morphisms that preserve divisible cells are equivalent to bicategories and (pseudo)functors; through a natural monoidal biclosed structure on merge-bicategories, one can also recover higher morphisms. I will use this to develop a semi-strictification argument for bicategories, combining the explicit combinatorial aspects of string-diagram based coherence proofs, with the main features of Hermida's abstract proof based on representable multicategories. All the constructions can be generalised to higher dimensions: I will sketch how this could lead to semi-strictification (excluding weak units) for higher categories, where it is still an open problem.