Why the category of all $n$-categories is a $(n+1)$-category?

Question

We have functors as $1$-morphisms, natural transformations as $2$-morphisms, but how the $k$-morphisms for $(n+1)\ge k\ge 3$ are built?

Answer 1

A morphism of $n$-categories is not really a functor, but an $n$-functor: the composition and identities have to be preserved for $k$ morphisms for all $1\leq k\leq n$. A 2-morphism of $n$-categories is similarly not just a natural transformation but an $n$-transformation. One has morphisms $F(x)\to G(x)$ for every $x$ satisfying a naturality condition, but also higher naturality conditions regarding the higher morphisms. A $3$-morphism of $n$-categories is called a modification, which you can see defined here.

Beyond this there aren't names, and few people consider the yet-higher morphisms explicitly. One needs a more uniform definition, which along the lines of another answer constructs the category of $n$-categories as a Cartesian closed category. This is well known for categories, and if one defines $n$-categories as categories enriched in $(n-1)$-categories, then abstract enriched category theory arguments show that $n$-categories are also Cartesian closed. In principle one can chase through the formalism to find an explicit definition of $k$-morphisms for $k>3$.

Answer 2

This is really not an answer to your question on how the $k>2$ cells are build but more a heuristic answer.

For $1$-categories the category of small categories is cartesian closed i.e. there exists and internal hom. These should really be seen as the category of functors between categories. In fact when $\mathrm{Fun}(C,D)$ is considered a category in the obvious way it models the internal hom in $\mathrm{Cat}$. Now for $n\in \mathbb{N}\cup\{\infty\}$, any model for $n$-categories we would want there to be an $n$-category of functors between objects in the $n$-category of $n$-categories $\mathrm{Cat}_n$.

Now here is the upshot in the $n$-category $\mathrm{Fun}_n(C,D)$ we would want the $0$-cells to be the $n$-functors, hence making $\mathrm{Cat}_n$ into an "$(n+1)$-category".