Short version: This is a reference request question. I would like to know if something has been written on the connection between $T$-multicategory (for $T$ a monad on a category $\mathcal{E}$), and categories $V \rightarrow \mathcal{E}_T$ over the Kleisli category of $T$ that are "patially fibered" in the sense that arrows in $\mathcal{E} \subset \mathcal{E}_T$ admits Cartesian lift.
Fore more details: Given a monad $T$ on a category $\mathcal{E}$ with finite limits, one can define the notion of $T$-multicategories, first considered by Burroni under the names $T$-categories in T-categories (Categories dans un Triple).
A $T$-categories is a pair of objects $(X_0,X_1)$ of $\mathcal{E}$ with maps $t:X_1 \rightarrow X_0$ and $s:X_1 \rightarrow T(X_0)$ and a composition map $T(X_1) \times_{T(X_0)} X_1 \rightarrow X_1$ satisfying some relation that recovers the usual notion of category when $T$ is the identity monad $Id$. The precise definition can be found in Burroni's paper linked above on page 1.1.
Most of the litterature on the topic has latter restricted to the case where $T$ is a Cartesian monad (or a club, or something similar), in which case one can give a slightly more conceptual definition (see for eg nLab), but the explicitly written definition actually make sense for any monad.
Now, given a (ordinary) category object $C=(C_0,C_1)$ in $\mathcal{E}$, one can construct a fibered category $C(\mathcal{E}) \rightarrow \mathcal{E}$ whose object are pairs $(e\in \mathcal{E}, c :e \rightarrow C_0)$ and morphisms $(e,c) \rightarrow (e',c')$ are pairs $(b:e \rightarrow e', f: e \rightarrow C_1)$ such that $t f= c$ and $ s f = c' b $. With some appropriate definition, this allows to show things along the line of:
Proposition: The 2-category of category objects in $\mathcal{E}$ is equivalent to the 2-category of internally essentially small fibered category over $\mathcal{E}$.
Proposition: The 1-category of category objects in $\mathcal{E}$ is equivalent to the category of internally small fibered category over $\mathcal{E}$ with explicitly chosen object of object.
I'm interested by an analogue of this construction for $T$-multicategory, I think I already have the correct statement:
Given a $T$-multicategory $X=(X_0,X_1)$ in $\mathcal{E}$ one can construct a category $X(\mathcal{E})$ over the Kleisli category $\mathcal{E}_T$ as follow:
Its objects are still pairs $(e \in \mathcal{E}, x:e \rightarrow X_0)$, and its morphisms $(e,x) \rightarrow (e',x')$ are pairs:
$$(b:e \rightarrow T e', f:e \rightarrow X_1)$$
such that $t f = x$ and $s f = T(x') b$. Composition being defined in the "obvious" way (not so obvious I agree, I mean the only reasonable thing to write down...).
This category $X(\mathcal{E}) \rightarrow \mathcal{E}_T$ is "partially fibered" in the sense that all arrows in $\mathcal{E} \subset \mathcal{E}_T$ admit cartesian lift, but not the general arrows of $\mathcal{E}_T$. I'm convince one can formulate and proove analogue of the two proposition above in this context of generalized multicategories. I would like to know if this has been done, or if something similar has been studied in the litterature somewhere.
An interesting example of this in the litterature, but that do not really explain the connection is this paper, were these partial fibration ("Weak Segal fibrations" in the paper) are used to give $\infty$-categorical definition of objects colored Operad etc... that are (in the $1$-categorical case) special case of $T$-Multicategories.
