# Fiber product in the category of DG categories

In Drinfeld’s paper DG quotient of DG categories 2.8 He says

Given DG functors $A’ \rightarrow A \leftarrow A’’$ one defines $A’ \times _A A’’$ to be the fiber product in the category of DG categories. This is the most naive definition(one takes the fiber product both at the level of objects and at the level of morphisms).

I am confused: a fiber product is defined to be some object together with some morphisms that is universal. But I can’t follow what Drinfeld said about the fiber product of DG categories.

$\require{AMScd}$In usual set-based categories you can define the objects of the fiber product of a diagram $$\begin{CD} @. C \\ @. @VVFV\\ D @>>G> X \end{CD}$$ taking the subset of the product of the classes of objects of $C\times D$ $$(C\times_XD)_0=\{(c,d)\mid Fc=Gd\}$$ and the hom-objects $(C\times_XD)[(x,a),(y,d)]$ are defined by doing the pullbacks of hom-sets $$\begin{CD} \bullet @>>> C(x,y) \\ @VVV \lrcorner @VVV\\ D(a,b) @>>> X(Fx,Fy) \end{CD}$$ This is how the proof that $\bf Cat$ is complete goes: you check that with this definition this category (you check that it is a category!) has the right universal prop.
Now, you can easily realize that the same thing shall give you the object with the right universal property in DG-categories too (do the second pullback in the category of chain complexes) or in any category $\cal V$ that is a "base" to do enriched category theory (at least a finitely complete and cocomplete, symmetric monoidal closed category).
• Thank you! I never met constructions like this before. At the level of objects, is it definitely $Fc=Gd$ or just $Fc\sim Gd$? I am afraid that this subset is empty. One more question: is this the most naive sense of product in the category of all (small) categories? Commented Dec 16, 2017 at 3:24