Let $G$ be a group and let $G$-$\textbf{Set}$ be the category of $G$-sets (whose objects are sets equipped with an action of $G$ and morphisms are G-equivariant maps). Determine The center of $G$-$\textbf{Set}$.
According to the definition, the elements of $Z(G\text{-}\textbf{Set})$ are automorphisms $\psi_X: X\to X$, such that the diagram
$X\overset{\psi_X}{\to} X$
$\!\!\!f\downarrow \quad\downarrow f $
$Y\overset{\psi_Y}{\to}Y$
is commutative for any $f: X\to Y$.
How to determine the center? Any hints are appreciated. Thanks in advance.