  In this paper, which is subsequent to our previous paper [PS] (but can be
read independently from it), we continue our study of the closed model
structure on the category $\mathrm{Cat}_{\mathrm{dgwu}}(\Bbbk)$ of small weakly
unital dg categories (in the sense of Kontsevich-Soibelman [KS]) over a field
$\Bbbk$. In [PS], we constructed a closed model structure on the category of
weakly unital dg categories, imposing a technical condition on the weakly
unital dg categories, saying that $\mathrm{id}_x\cdot
\mathrm{id}_x=\mathrm{id}_x$ for any object $x$. Although this condition led us
to a great simplification, it was redundant and had to be dropped.
  Here we get rid of this condition, and provide a closed model structure in
full generality. The new closed model category is as well cofibrantly
generated, and it is proven to be Quillen equivalent to the closed model
category $\mathrm{Cat}_\mathrm{dg}(\Bbbk)$ of (strictly unital) dg categories
over $\Bbbk$, given by Tabuada [Tab1]. Dropping the condition
$\mathrm{id}_x^2=\mathrm{id}_x$ makes the construction of the closed model
structure more distant from loc.cit., and requires new constructions. One of
them is a pre-triangulated hull of a wu dg category, which in turn is shown to
be a wu dg category as well.
  One example of a weakly unital dg category which naturally appears is the
bar-cobar resolution of a dg category. We supply this paper with a refinement
of the classical bar-cobar resolution of a unital dg category which is strictly
unital (appendix B). A similar construction can be applied to constructing a
cofibrant resolution in $\mathrm{Cat}_\mathrm{dgwu}(\Bbbk)$.
