I have been reading Gonçalo Tabuada's paper Higher K-theory via universal invariants in a seminar and the following question arose.
At one point in his construction (specifically $\S$10) he looks at the set of full dg-inclusions $\mathcal G\hookrightarrow\mathcal H$ where $\mathcal H$ is a strict finite $I$-cell, where $I$ is the usual (I suppose) set of generating cofibrations of the category dgcat of small dg-categories where we take Morita equivalences as the weak equivalences. The interest is in the Drinfeld quotient $\mathcal H\,/\,\mathcal G$.
The main question I have is the following: is this quotient still (weakly/Morita equivalent to) a strict finite $I$-cell? I'm not sure this is required for Tabuada's argument to hold, but it would be nice to know if we can take the quotient in the 'finite' part of dgcat.