# Collages along composition of distributors

The construction of a collage of two categories $\bf A,B$ along a profunctor $\phi\colon \bf A\mid\hspace{-2mm}\to B$ gives a new category $\bf A \uplus_\phi B$ having as objects those of $\bf A\amalg B$, arrows $A\to A'$ are $\mathbf A(A,A')$, arrows $B\to B'$ are $\mathbf B(B,B')$, $A\to B$ are $\phi(A,B)$ and $\mathbf A \uplus_\phi\mathbf B(B,A)=\varnothing$.

Now I'm wondering if given two distributors $\phi\colon \mathbf A\mid\hspace{-2mm}\to \mathbf B$, $\psi\colon \mathbf B\mid\hspace{-2mm}\to\mathbf C$ there is a sensible relation between $\mathbf A\uplus_{\psi\circ\phi}\mathbf C$ and $\mathbf A\uplus_\phi\mathbf B$, $\mathbf B\uplus_\psi \mathbf C$.

In the setting of categories enriched in a bicategory, Carboni, Kasangian, and Walters define collages as `colimits' of $\bf Prof$-categories. The collage of a profunctor $\phi \colon \mathbf{A} \mid\hspace{-2.2mm}\to \mathbf{B}$ is then the colimit of the $\bf Prof$-category $\mathbb{C}_\phi$ with two objects $a$ and $b$ of respective types $\bf A$ and $\bf B$, with $\mathbb{C}_\phi(a,a) = \mathit{id}_{\bf A}$, $\mathbb{C}_\phi(b,b) = \mathit{id}_{\bf B}$, $\mathbb{C}_\phi(b,a) =$ the empty profunctor, and $\mathbb{C}_\phi(a,b) = \phi$.
One $\bf Prof$-category that you haven't considered is, say $\mathbb{C}_{\phi,\psi}$, which has three objects $a,b,c$ of types $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$, with obvious hom-profunctors. There are inclusion $\bf Prof$-functors from $\mathbb{C}_{\phi}$, $\mathbb{C}_{\psi}$, and $\mathbb{C}_{\psi \circ \phi}$, into $\mathbb{C}_{\phi,\psi}$, which induce proper functors between collages.