Let $X,Y,Z$ be categories and

$D:X^{op}\times Y\to Cat$ and

$E:Y^{op}\times Z\to Cat$

be $Cat$-valued distributors; (strong/pseudo)-functors.

Their composite can then be described as a coend or as a coequalizer. The corresponding lax- or oplax colimits can be calculated as grothendieck constructions on the category with two parallel arrows. But what about the strong coend?

Question 1: Is there a nice and explicit description of the composite?

The reason i am asking is that the grothendieck construction for a distributor gives a 2-equivalence between between $Cat$-vaulued distributors and spans that are fibrations on one side and opfibrations on the other. My first idea of showing this would be by using the explicit description of the grothendieck construction and some calculations.

On the other hand, such spans can be described as spans with left and right actions of the morphism-double-categories respectively (they are 2-monoids in the 2-category of endo-spans). Now i have a formula in mind that should look like

$$\int D\otimes_Y E = \left(\int D\right) \times_Y \left(\int E\right)$$

(Here $\times_Y$ denotes the strong pullback).

Then - using $\int hom = Mor$ - we would get

$$\int hom\otimes_X E = Mor_X \times_X \int E$$

and the operation

$$hom\otimes_X E \to E$$

should directly induce an operation

$$Mor_X\times_X \int E \to E$$

that exposes $\int E$ as fibered over $X$.

Question 2: What's the story here? Is my guess correct? And can this be shown without reference to the explicit description of grothendieck construction?

Remark: "$=$" is of course to be read as an equivalence.

EDIT: I'm posting another related question; link coming soon.