Lie correspondence and functor Lie

Question

I am looking for a simple proof regarding the fact that the assignmaent $Lie:G \rightarrow Lie(G)$ is functor from $LG$ the category of Lie groups to the category $LA$ of Lie algebras.

In fact I'm looking for any information about the Lie correspondence,for my final exam. Can you help me please?

Answer 1

this is just differentiation at the unitary of G. and also with Lie Deriviative (derivative of derivative of conjugate representation) you have the [,] product over Lie(G).