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

A closed model structure on the category of weakly unital dg categories, II