  There are known two different constructions of contractible dg 2-operads,
providing a weak 2-category structure on the following dg 2-quiver of small dg
2-categories. Its vertices are small dg 2-categories over a given field, arrows
are dg functors, and the 2-arrows $F\Rightarrow G$ are defined as the
Hochschild cochains of $C$ with coefficients in $C$-bimodule $D(F(-),G(=))$,
where $F,G\colon C\to D$ are dg functors, $C,D$ small dg categories. It is
known that such definition is correct homotopically, but, on the other hand,
the corresponding dg 2-quiver fails to be a strict 2-category.
  The question ``What do dg categories form'' is the question of finding a weak
2-category structure on it, in an appropriate sense. One way of phrasing it out
is to make it an algebra over a contractible 2-operad, in the sense of
M.Batanin [Ba1,2] (in turn, there are many compositions of 2-arrows for a given
diagram, but their totality forms a contractible complex) .
  In [T], D.Tamarkin proposed a contractible $\Delta$-colored 2-operad in Sets,
whose dg condensation solves the problem. In our recent paper arXiv:1807.04305
we constructed contractible dg 2-operad, called the twisted tensor product
operad, acting on the same 2-quiver (the construction uses the twisted tensor
product of small dg categories introduced in arXiv:1803.01191).
  In this paper, we compare the two constructions.
