Berger's wreath construction as adjoint to taking cocategories?

Is the following an adjuntion? $\Delta \wr (-) : Cat^{po} \rightleftarrows Cat_\ast^{po} : coCat_*(-)$

where

• $Cat^{po}$ is the category of realized pushout sketches (i.e. an object is a pair $(\mathcal C, J)$ where $\mathcal C$ is a small category and $J$ is a set of pushout squares in $C$, a morphism is a functor preserving sending designated pushout squares to designated pushout squares. I think we should also require that any commutative square of identity morphisms is "designated".)
• $Cat^{po}_\ast$ is the category of pointed objects in $Cat^{po}$ (equivalently, it's the coslice category $Cat^{po}_{\ast / }$ where $\ast = (\{pt\}, \{const_{pt}\})$ is the terminal category with the unique pushout square is "designated")
• $\Delta \wr (-)$ is Berger's wreath construction, where we take $[0]$ as our pointing.
• $coCat_\ast(-)$ takes cocategory objects (where a cocategory object is taken to be a cosimplicial objects satisfying coSegal conditions (saying that certain squares are pushouts -- which we require to be among the "designated" pushouts)

I believe the answer is yes (or nearly so! see caveat below!), and might be in the literature (perhaps the "categorical pattern" literature à la Barwick, Lurie, Haugseng,...).

Note that in (1,1)-categories, you can get away with defining a cocategory object via a 2-truncated cosimplicial diagram. This should correspond to a version of Berger's construction using $\Delta_{\leq [2]}$ instead of $\Delta$.

It's clear that $coCat_\ast : Cat^{po}_\ast \to Cat^{po}$ is corepresentable by $([0] \in (\Delta , \{\text{cosegal pushouts}\}))$, using the obvious self-enrichment of $Cat^{po}$, which lifts to $Cat^{po}_\ast$. So the left adjoint $L$ to $coCat_\ast$ exists, and there's even a formula for it: we have $L(\mathcal C , J) = \Delta \otimes J \cup_J \ast$ where $\otimes$ is adjoint to the internal hom for $Cat^{po}$ (at the level of categories, this is the cartesian product), and the pushout is taken in $Cat^{po}$. The issue is that it's a bit delicate to compute pushouts in $Cat^{po}$, in order to verify that $L \cong \Delta \wr (-)$ (notably, propagating the condition that the designated squares be pushouts affects the hom-sets). The quotient map $\Delta \otimes \mathcal C \to \Delta \wr \mathcal C$ presumably will send $([n], C) \mapsto ([n]; C, \dots, C)$.

Caveat: It's possible that the way I've described it, we should actually get the full subcategory of $\Delta \wr \mathcal C$ on objects of the form $([n]; C, C, \dots, C)$. In cases like $\mathcal C = \Theta_n$, for any $C_1, \dots C_n \in \mathcal C$ there is some $C \in \mathcal C$ such that each $C_i$ is a retract of $C$. Then $([n]; C_1, \dots, C_n)$ is a retract of $([n]; C, \dots, C)$, so these two constructions should be the same up to idempotent completion at least...

Sorting out such subtleties is another reason to hope that some discussion of this adjunction appears in the literature already!