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This is very transparent when using the definition $$ Hom_{Cat}(\mathcal C,\mathcal D):=Hom_{t_{\leq0}(Cat)}(\mathcal C\times\Delta^1,\mathcal D)\,, $$ where $\Delta^1=\Box^1=G^1$ is the arrow category $0\to 1$ and $t_{\leq0}(Cat)$ is the underlying $1$-category of $Cat$.

Le us now go to natural transformations of (strict) $2$-functors between (strict) $2$-categories. Given two such $2$-functors $F,G:\mathcal C\to\mathcal D$ one can see that a natural transformation $F\Rightarrow G$ is the same as a $2$-functors $$ \phi:\mathcal C\times G^2\to\mathcal D $$ such that $\phi(-,0)=F$ and $\phi(-,0)=G$, where $G^2$ is the $2$-category with two objects $0$ and $1$ and such that $Hom_{G^2}(0,1)$ is the arrow category $G^1=(0\to 1)$.

The globe category $G$, the cubical category $\Box$ and the simplicial category $\Delta$ are known to be suitable geometric shape to model higher structures. Simplicial sets are good models for (weak) $\infty$-groupoids. It was proved (by Jardine ... with some improvement by Cisinski if I remember well) that cubical sets also provide a model for (weak) $\infty$-groupoids.

Let me consider the category $n-Cat$ of (strict) $n$-categories. A a natural transformation between (strict) $n$-functor $F,G:\mathcal C\to\mathcal D$ can be written as an $n$-functor $$ \phi:\mathcal C\times G^n\to\mathcal D $$ such that $\phi(-,0)=F$ and $\phi(-,0)=G$, where $G^n$ is the $n$-category with two objects $0$ and $1$ and such that $Hom_{G^n}(0,1)$ is the $(n-1)$-category $G^{n-1}$.

This is very transparent when using the definition $$ Hom_{Cat}(\mathcal C,\mathcal D):=Hom_{t_{\leq0}(Cat)}(\mathcal C\times\Delta^1,\mathcal D)\,, $$ where $\Delta^1=\Box^1=\mathbb{G}^1$ is the arrow category $0\to 1$ and $t_{\leq0}(Cat)$ is the underlying $1$-category of $Cat$.

Le us now go to natural transformations of (strict) $2$-functors between (strict) $2$-categories. Given two such $2$-functors $F,G:\mathcal C\to\mathcal D$ one can see that a natural transformation $F\Rightarrow G$ is the same as a $2$-functors $$ \phi:\mathcal C\times \mathbb{G}^2\to\mathcal D $$ such that $\phi(-,0)=F$ and $\phi(-,1)=G$, where $\mathbb{G}^2$ is the $2$-category with two objects $0$ and $1$ and such that $Hom_{\mathbb{G}^2}(0,1)$ is the arrow category $\mathbb{G}^1=(0\to 1)$.

The globe category $\mathbb{G}$, the cubical category $\Box$ and the simplicial category $\Delta$ are known to be suitable geometric shape to model higher structures. Simplicial sets are good models for (weak) $\infty$-groupoids. It was proved (by Jardine ... with some improvement by Cisinski if I remember well) that cubical sets also provide a model for (weak) $\infty$-groupoids.

Let me consider the category $n-Cat$ of (strict) $n$-categories. A a natural transformation between (strict) $n$-functor $F,G:\mathcal C\to\mathcal D$ can be written as an $n$-functor $$ \phi:\mathcal C\times \mathbb{G}^n\to\mathcal D $$ such that $\phi(-,0)=F$ and $\phi(-,1)=G$, where $\mathbb{G}^n$ is the $n$-category with two objects $0$ and $1$ and such that $Hom_{\mathbb{G}^n}(0,1)$ is the $(n-1)$-category $\mathbb{G}^{n-1}$.

Let me consider the category $n-Cat$ of (strict) $n$-categories. A (strict) $n$-functor can be written as a functor $$ \phi:\mathcal C\times G^n\to\mathcal D $$ such that $\phi(-,0)=F$ and $\phi(-,0)=G$, where $G^n$ is the $n$-category with two objects $0$ and $1$ and such that $Hom_{G^n}(0,1)$ is the $(n-1)$-category G^{n-1}$.

Let me consider the category $n-Cat$ of (strict) $n$-categories. A a natural transformation between (strict) $n$-functor $F,G:\mathcal C\to\mathcal D$ can be written as an $n$-functor $$ \phi:\mathcal C\times G^n\to\mathcal D $$ such that $\phi(-,0)=F$ and $\phi(-,0)=G$, where $G^n$ is the $n$-category with two objects $0$ and $1$ and such that $Hom_{G^n}(0,1)$ is the $(n-1)$-category $G^{n-1}$.

This is very transparent using the definition $$ Hom_{Cat}(\mathcal C,\mathcal D):=Hom_{t_{\geq1}(Cat)}(\mathcal C\times\Delta^1,\mathcal D)\,, $$ where $\Delta^1=\Box^1=G^1$ is the arrow category $0\to 1$ and $t_{\geq 1}(Cat)$ is the underlying $1$-category of $Cat$.

Remark: In general one can see a $2$-category $\mathcal C$ as a simplicial category replacing the $Hom$-categories by their nerves.

In the case of $Cat$, we see that the $Hom$-category naturally appear as $1$-truncations of simplicial sets (one can replace here "simplicial" by "cubical" of "globular").

Remark: as before we can then see any $3$-category as a simplicial/cubical/globular category by replacing the $Hom$-$2$-categories by their (simplicial/cubical/globular) nerves.

In the case of $2-Cat$, we see that the $Hom$-$2$-category naturally appear as $2$-truncations of globular sets.

This is very transparent when using the definition $$ Hom_{Cat}(\mathcal C,\mathcal D):=Hom_{t_{\leq0}(Cat)}(\mathcal C\times\Delta^1,\mathcal D)\,, $$ where $\Delta^1=\Box^1=G^1$ is the arrow category $0\to 1$ and $t_{\leq0}(Cat)$ is the underlying $1$-category of $Cat$.

Remark: In general one can see a $2$-category $\mathcal C$ as a simplicial category by replacing the $Hom$-categories by their nerves.

In the case of $Cat$, we see that the $Hom$-categories naturally appear as $1$-truncations of simplicial sets (one can replace here "simplicial" by "cubical" of "globular").

Remark: as before we can then see any $3$-category as a simplicial/cubical/globular category by replacing the $Hom$-$2$-categories by their (simplicial/cubical/globular) nerves.

In the case of $2-Cat$, we see that the $Hom$-$2$-categories naturally appear as $2$-truncations of globular sets.