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 $\mathbf A\uplus_\phi\mathbf B$, $\mathbf B\uplus_\psi \mathbf C$.