Let $G$ be a lie group.
Let $$C: G\times G \longrightarrow G$$ $$C(g,h)=ghg^{-1}$$
I have proven that the left $L_g$ and right $R_g$ multiplications are smooth. Therefore $$C_g: G \longrightarrow G$$ $$C_g(h)=ghg^{-1}$$
But how do I use these facts to prove that $C$ is smooth? I'm trying to write it as the composition of smooth functions but I am having trouble doing so.
The map $\phi:G\times G\rightarrow G, (g,h)\mapsto g^{-1}$ is smooth, since it is the projection onto the first factor composed with the inverse map, which are both smooth. Thus $G\times G\rightarrow G\times G, (g,h)\mapsto (gh,g^{-1})$ is smooth, as it is the product of $\phi$ and the multiplication map. Applying the multiplication map again gives the result.