Now consider a $G$-set $X \colon \mathsf{B} G \to \textup{\textsf{cat}}$. Example \ref{exs:natural}\eqref{itm:equivariant} observed that a natural transformation $\phi \colon G \Rightarrow X$ is exactly a $G$-equivariant map $\phi \colon G \to X$. Here $\phi \colon G \to X$ is the unique component of the natural transformation, and equivariance of this map expresses the naturality condition. To define $\phi$ we must specify elements $\phi(g) \in X$ for each $g \in G$. Equivariance demands that $\phi(g \cdot h) = g \cdot \phi(h)$. Taking $h$ to be the identity element, we see that $\phi(g) = g \cdot \phi(e)$. In other words, the choice of $\phi(e) \in X$ forces us to define $\phi(g)$ to be $g \cdot \phi(e)$. Moreover, any choice of $\phi(e) \in X$ is permitted, because the left action of $G$ on $G$ is free.