This is a particular case of this question.
For any categories $\mathcal C$ and $\mathcal D$, write $\mathcal D^{\mathcal C}$ for the category of functors from $\mathcal C$ to $\mathcal D$.
Set $\mathcal C:=\mathsf{Set}^{\mathsf{Set}}$.
Is $\mathcal C^{\mathcal C}$ equivalent to $\mathcal C$?
Note that $\mathsf{Set}^{\mathsf{Set}}$ is not equivalent to $\mathsf{Set}$. Indeed, for any category $\mathcal A$ with terminal object $1$ and coproduct $2:=1\sqcup1$, and any object $A$ of $\mathcal A$, write $P(A)$ for the statement "there are exactly two morphisms from $A$ to $2$ in $\mathcal A$". In $\mathsf{Set}$ the condition $P(X)$ implies $X\simeq1$, whereas in $\mathsf{Set}^{\mathsf{Set}}$ any representable functor $F$ satisfies $P(F)$ by the Yoneda Lemma. This is easily seen to imply $\mathsf{Set}^{\mathsf{Set}}\not\sim\mathsf{Set}$.