  It is well-known that the "pre-2-category"
$\mathscr{C}at_\mathrm{dg}^\mathrm{coh}(k)$ of small dg categories over a field
$k$, with 1-morphisms defined as dg functors, and with 2-morphisms defined as
the complexes of coherent natural transformations, fails to be a strict
2-category. In [T2], D.Tamarkin constructed a contractible 2-operad in the
sense of M.Batanin [Ba3], acting on
$\mathscr{C}at_\mathrm{dg}^\mathrm{coh}(k)$. According to Batanin loc.cit., it
is a possible way to define a "weak 2-category".
  In this paper, we provide a construction of {\it another} contractible
2-operad $\mathcal{O}$, acting on $\mathscr{C}at_\mathrm{dg}^\mathrm{coh}(k)$.
Our main tool is the {\it twisted tensor product} of small dg categories,
introduced in [Sh3]. We establish a one-side associativity for the twisted
tensor product, making
$(\mathscr{C}at_\mathrm{dg}^\mathrm{coh}(k),\overset{\sim}{\otimes})$ a skew
monoidal category in the sense of [LS], and construct a {\it twisted
composition}
$\mathscr{C}oh_\mathrm{dg}(D,E)\overset{\sim}{\otimes}\mathscr{C}oh_\mathrm{dg}(C,D)\to\mathscr{C}oh_\mathrm{dg}(C,E)$,
and prove some compatibility between these two structures. Taken together, the
two structures give rise to a 2-operad $\mathcal{O}$, acting on
$\mathscr{C}at_\mathrm{dg}^\mathrm{coh}(k)$. Its contractibility is a
consequence of a general result of [Sh3].
