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It is a well-known result (by G. Tabuada, as I recall) that the category $\mathbf{dgcat}_k$ of (small) dg-categories over a fixed commutative ring $k$ admits a model category structure such that the weak equivalences are the quasi-equivalences. Then, necessarily, $\mathbf{dgcat}_k$ is small complete and cocomplete, in particular it admits pullbacks and pushouts. However, I could not find any description of them. Actually, I don't even know what should be the product (or coproduct) of two dg-categories, nor could I find any kind of reference.

Do you have any clue?

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