Garner constructed (in [[1]](https://arxiv.org/abs/0810.4450) ) for the category of strict $n$-categories a comonad $Q$ as the left part of a cofibrantly generated algebraic weak factorization system such that $$n\text{-}\operatorname{Cat}(QX,Y)=\operatorname{Pseudo}(X,Y).$$ 


For various reasons, it is often more convenient to work with *normalized* pseudofunctors, that is, pseudofunctors preserving units strictly. Steve Lack and Simona Paoli gave a construction of such a comonad in the case $n=2$ implicitly in [[2]](https://arxiv.org/abs/math/0607271).


Is there any variant of Garner's construction, hopefully as part of a cofibrantly generated algebraic weak factorization system adapted to the folk model structure on strict $n$-categories that corepresents the normalized pseudofunctors in the same way?



Comonad for normalized pseudofunctors for strict higher categories
$\require{AMScd}$Notation: for each $n \geq 0$, let $\mathbf{2}\_n$ denote the free-living $n$-cell, and let $\partial\mathbf{2}\_n$ denote its boundary. Let **$n$-Cat** denote the category of (strict) $n$-categories and (strict) $n$-functors.


Recall (see e.g. Section 7.2 of Garner's `Understanding the small object argument') that the "pseudofunctor classifier" (or "strictification") comonad on the category **2-Cat** is the cofibrant replacement comonad of the awfs (= algebraic weak factorisation system) on **2-Cat** generated (via the algebraic small object argument) by the set of 2-functors $$\{\partial\mathbf{2}\_0 \to \mathbf{2}\_0, \partial\mathbf{2}\_1 \to \mathbf{2}\_1, \partial\mathbf{2}\_2 \to \mathbf{2}\_2, \partial\mathbf{2}\_3 \to \mathbf{2}\_2\}.$$


One can similarly obtain the "*normal* pseudofunctor classifier" (or "normal strictification") comonad on **2-Cat** as the cofibrant replacement comonad for the awfs on **2-Cat** generated by a certain *category* $\mathcal{J}\_2$ of 2-functors (i.e. a subcategory of the arrow category of **2-Cat**). The objects of this category $\mathcal{J}\_2$ are the 2-functors listed above, together with the identity 2-functor $\mathbf{2}\_0 \to \mathbf{2}\_0$; the only non-identity morphism in $\mathcal{J}\_2$ is the unique morphism in the arrow category of **2-Cat** from $\partial\mathbf{2}\_1 \to \mathbf{2}\_1$ to $\mathbf{2}\_0 \to \mathbf{2}\_0$, i.e. the following commutative square of 2-functors. 
\begin{CD}
 \partial\mathbf{2}\_1 \ @>>> \mathbf{2}\_0\\
 @V V V @VV V\\
 \mathbf{2}\_1 @>>> \mathbf{2}\_0
\end{CD}
(Intuitively, the effect of this change to the algebraic small object argument is that a cell is attached only for each "non-degenerate" lifting problem.)


The natural generalisation of this construction would be to define the "normal pseudofunctor classifier" comonad on **$n$-Cat** to be the cofibrant replacement comonad for the awfs on **$n$-Cat** generated by the category $\mathcal{J}\_n$ of $n$-functors whose objects are the following "boundary inclusions" and identity $n$-functors:
$$\{\partial\mathbf{2}\_0 \to \mathbf{2}\_0, \ldots, \partial\mathbf{2}\_n \to \mathbf{2}\_n, \partial\mathbf{2}\_{n+1} \to \mathbf{2}\_n\}\cup\{\mathbf{2}\_0 \to \mathbf{2}\_0, \ldots, \mathbf{2}\_{n-1} \to \mathbf{2}\_{n-1}\},$$ 
and whose only non-identity morphisms are the commutative squares
\begin{CD}
 \partial\mathbf{2}\_{k+1} \ @>>> \mathbf{2}\_k\\
 @V V V @VV V\\
 \mathbf{2}\_{k+1} @>>> \mathbf{2}\_k
\end{CD}
(where the bottom $n$-functor picks out the identity $(k+1)$-cell of the non-identity $k$-cell of $\mathbf{2}\_k$) for each $0 < k < n$. 
If we let $Q$ denote the comonad so constructed, this generalisation suggests that a *normal pseudofunctor* between $n$-categories $A \rightsquigarrow B$ is a strict $n$-functor $QA \to B$. 


