ScalingStacks

Proof. Such a natural transformation consists of component maps (i.e., functors) $\eta_{X}\colon X\to X$ for each gaunt $n$-category $X$. We will show that $\eta_{X}=\id_{X}$ for all $X$. The functor $\eta_{X}$ induces, for each $0\leq k\leq n$, a map on sets of $k$-cells,

$$(\eta_{X})_{k}\colon X_{k}\to X_{k}.$$

Since a functor is completely determined by the map on $k$-cells for each $k$, it is enough to show that each $(\eta_{X})_{k}$ is the identity. By naturality of $\eta$ it is enough to show that the single functor $\eta_{C_{n}}=\id_{C_{n}}$.

One my now show that $\eta_{C_{k}}=\id_{C_{k}}$ by inducting on $k$. When $k=0$ the claim is obvious. Now the inductive hypothesis asserts $\eta_{C_{k}}$ is a functor which restricts to the identity functor on $\partial C_{k}$. There is only one functor with this property, namely $\eta_{C_{k}}=\id_{C_{k}}$. ∎