  Poly-bicategories generalise planar polycategories in the same way as
bicategories generalise monoidal categories. In a poly-bicategory, the
existence of enough 2-cells satisfying certain universal properties
(representability) induces coherent algebraic structure on the 2-graph of
single-input, single-output 2-cells. A special case of this theory was used by
Hermida to produce a proof of strictification for bicategories. No full
strictification is possible for higher-dimensional categories, seemingly due to
problems with 2-cells that have degenerate boundaries; it was conjectured by C.
Simpson that semi-strictification excluding units may be possible.
  We study poly-bicategories where 2-cells with degenerate boundaries are
barred, and show that we can recover the structure of a bicategory through a
different construction of weak units. We prove that the existence of these
units is equivalent to the existence of 1-cells satisfying lower-dimensional
universal properties, and study the relation between preservation of units and
universal cells.
  Then, we introduce merge-bicategories, a variant of poly-bicategories with
more composition operations, which admits a natural monoidal closed structure,
giving access to higher morphisms. We derive equivalences between morphisms,
transformations, and modifications of representable merge-bicategories and the
corresponding notions for bicategories. Finally, we prove a
semi-strictification theorem for representable merge-bicategories with a choice
of composites and units.
