Suppose $M$ is a compact $n$-dimensional manifold without boundary. Let $H^s(M,M)$ denote the Sobolev space on $M$, defined as all maps from $M$ to $M$ whose distributional derivatives up to order $s$ are square integrable. For $s>n/2+1$, let $$\mathcal{D}^s =\{\eta\in H^s(M, M)| \eta ~\text{is bijective and}~ \eta^{-1}\in H^s(M,M)\}. $$ Define right multiplication \begin{align} R_\eta:\mathcal{D}^s \to\mathcal{D}^s \\ \xi\to\xi\circ\eta. \end{align} Why is $R_\eta ~C^{\infty}$ for each $\eta \in \mathcal{D}^s$?

Also, if define left multiplication \begin{align} L_\eta :\mathcal{D}^s \to\mathcal{D}^s \\ \xi \to \eta \circ\xi. \end{align} Why is $L_\eta$ only $C^l$ when $\eta\in \mathcal{D}^{s+l}$?

Essentially what is the difference between $R_\eta ~\text{and} ~L_\eta$?