In some recent works, such as this one (3.2, page 15), a definition of “gluing of dg-categories along a dg-bimodule” is given. It is obviously the analogue of the notion of collage (or cograph) of a profunctor.

My question is: is there any “universal property” of this gluing? Something like “a dg-functor defined on the gluing is uniquely determined by something defined on the “pieces” (the dg-categories and the bimodule)”. The article on ncat about the collage of a profunctor suggests the existence of a universal property, but I’m unable to write it down in a “honest” (hands-on) manner. Is there a way to do this in the world of dg-categories?

**Answer**

I think it is more natural to ask for a universal property with respect to quasifunctors. There is one and you can find it in Appendix A of http://arxiv.org/abs/1212.6170.

**Attribution***Source : Link , Question Author : Francesco Genovese , Answer Author : Sasha*