What is meant by the "functoriality of the adjoint representation"?

Solution 1:

What is meant here is that the adjoint representation is nicely compatible with homomorphisms of Lie groups. For a homomorphism $\phi:H\to G$, let $\phi':\mathfrak h\to\mathfrak g$ be the derivative. Then it is a basic fact of Lie theory that $\phi(exp(tX))=\exp(t\phi'(X))$ for all $X\in\mathfrak h$. Now for $h\in H$ and $X\in\mathfrak h$, you get Ad$(h)(X)$ as the derivative at $t=0$ of the curve $h\exp(tX)h^{-1}$. Using the above fact, you get $\phi(h\exp(tX)h^{-1})=\phi(h)\exp(t\phi'(X))\phi(h)^{-1}$. Differentiating at $t=0$, we obtain $\phi'(Ad(h)(X))=Ad(\phi(h))(\phi'(X))$, which is the functorial property the argument refers to. You just have to apply this to the inclusion $i:H\to G$ whose derivative $i'$ is the inclusion $\mathfrak h\hookrightarrow\mathfrak g$.