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$\require{AMScd}$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 set of objects is the union of the "boundary inclusions" and the 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 $A \rightsquigarrow B$ could be defined as a strict $n$-functor $QA \to B$.

$\require{AMScd}$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 $QA \to B$.

$\require{AMScd}$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 we only attach the cells corresponding to non-degenerate lifting problems.)

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 set of objects is the union of the "boundary inclusions" and the 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 $0 < k < n$.

$\require{AMScd}$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 set of objects is the union of the "boundary inclusions" and the 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 $A \rightsquigarrow B$ could be defined as a strict $n$-functor $QA \to B$.

$\require{AMScd}$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 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 (using 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 we only attach the cells corresponding to non-degenerate lifting problems.)

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 set of objects is the union of the "boundary inclusions" and the 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 $0 \leq k < n$.

$\require{AMScd}$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 we only attach the cells corresponding to non-degenerate lifting problems.)

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 set of objects is the union of the "boundary inclusions" and the 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 $0 < k < n$.