Universal property of gluing [collage, cograph] of dg-categories 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?
 A: 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.
A: $\newcommand{\nto}{\looparrowright}$ Collage along the terminal profunctor has, indeed, a universal property since it can be characterized via an adjunction.
But let me expand a little bit a piece of theory I never found written anywhere. This should turn into something extremely boring for skilled readers, so let me apologize in advance for my naivete.
I'll denote as $W\colon C\nto D$ a profunctor $W\colon C^{op}\times D\to \rm Set$. (generalizing to the enriched case is a matter of changing Set with a generic $\cal V$).
Fact. The collage operation $(C,D,W\colon C \nto D)\mapsto C\star_W D$ is functorial if we consider the category $\cal C$ having as objects triples $(C,D,W)$ where $C,D\in{\rm Cat}$ and $W\colon C\nto D$ is a profunctor.
This category is (a particular presentation of?) the proarrow equipment $({\rm Cat},\rm Dist)$.
Proof. $\cal C$ becomes a category if I define arrows $(C,D,W)\to (C',D', W')$ to be triples $(f,g,\eta)$ where


*

*$f\colon C\to C'$ and $g\colon D\to D'$ are functors;

*$\eta\colon W\to W'\diamond \phi^f\diamond \phi_g$ is a natural transformation: $\diamond$ is the composition of profunctors defined as a coend, $\phi^f=\hom(f,1),  \phi_g =\hom(1,g)$ are the ``representable'' profunctors.


(this is precisely what you need to define the equipment).
It's rather easy to see that this is a category: given two arrows $(f,g,\eta)$, $(f', g',\psi)$ composition is defined by
$$
W\xrightarrow{\eta} W'\phi^f\phi_g \xrightarrow{\psi \diamond \phi^f\diamond\phi_g} W''\phi^{f'}\phi_{g'}\phi^f\phi_g \cong W''\diamond\phi^{f'f}\diamond\phi_{g'g}
$$
(since it is obvious that $\phi^f\diamond \phi_g\cong \phi_g\diamond \phi^f$) and the identity arrow is the obvious one since $\phi^\text{id}=\phi_\text{id}=\hom$. We now have to prove that this defines a functor $C\star_W D\to C'\star_{W'}D'$ for any morphism $(f,g,\eta)$; this definition obviously collapses to the classical one if we consider $W,W'$ to be the terminal profunctors.
To this end, let's consider that a pair of functors $F\colon C\to C', G\colon D\to D'$ in the terminal case induced a functor $F\star G\colon C\star D\to C'\star D'$ (this can be proved fixing a component and letting the second vary, as the interchange law obviously holds); $F\star G$ is defined to be $F$ (on objects and arrows) when restricted to $C$, $G$ when restricted to $D$, and the unique arrow $c\to d$ is sent to the unique arrow $Fc\to Gd$. 

In the general case, there is no preferred choice for an arrow linking $\hom_{C\star_W D}(c,d)$ to $\hom_{C'\star_W D'}(Fc,Gd)$, so the assignment of $F\star_W G$ on arrows has to be made via a morphism between profunctors: this is the role of $\{ \eta_{c,d}\colon W(c,d)\to W'\diamond \phi^f\diamond \phi_g(c,d)\}$.
$F\star_W G$ is defined as $F$ on $C$, as $G$ on $D$, and the ``new-born'' arrows represented by $W(c,d)$ are sent to the arrows $W'(Fc,Gd)$ via $\eta$. Functoriality of $F,G$ and naturality of $\eta$ ensure that everything goes right. $\blacksquare$

Now for your real question: I think the popularization of this result is due to Joyal, or at least, I learned it in simplicial flavour through his notes (see chap. 3) about quategories, and reading his catlab.
$\def\Cat{\text{Cat}}$ Consider the inclusion of the boundary of the standard 1-simplex, $i\colon \{0,1\}\to [1]$ as a functor between the discrete category with two elements and the ``walking arrow'' $I=\{0 \leq 1\}$. It induces a functor
$$
i^*\colon \Cat / I 
\to  
\Cat \times \Cat
$$
which admits a right adjoint. This right adjoint is precisely the bifunctor $\star\colon \Cat \times \Cat \to \Cat / I$, given by the join along the terminal profunctor. 
Notice that this collapsing has precisely the effect of removing the obstruction for $(-)\star_{(-)}(-)$ to become "monoidally-flavoured".
The result is clear, once we noticed that the category $C\star C'$ comes naturally equipped with an arrow $C\star C'\to I=1\star 1$ induced by (bi)functoriality of $\star$, starting from the canonical arrows $C\to 1, C'\to 1$ to the terminal category: more precisely, it is clear that $i^*$ is defined by sending $C\to I$ to the pair of categories $i^\leftarrow(0)=C_0, C_1=i^\leftarrow(1)$. The bijection
$$
\Cat^{\mathbf{2}}\Big(i^*\big( C\to I\big), (A,B)\Big)\cong 
\Cat / I \Big( C, A\star B \Big)
$$
is now rather obvious, since any functor $i^*\big( C\to I\big)\to (A,B)$ determines a functor $C\to A\star B$ and viceversa. $\blacksquare$
Remark. If I remember well, $(\Cat, \star)$ is a fairly blatant example of a non-closed, biclosed (see Joyal 3.1.1, 3.1.2) monoidal category.
