lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -205,10 +205,10 @@ this last construction.
 
 It is important to point out that the construction of the Lie algebra
 \(\mathfrak{g}\) of a Lie group \(G\) in example~\ref{ex:lie-alg-of-lie-grp} is
-functorial. Specifically, one can show the derivative \(f^* : \mathfrak{g}
+functorial. Specifically, one can show the derivative \(d f_1 : \mathfrak{g}
 \cong T_1 G \to T_1 H \cong \mathfrak{h}\) of a smooth group homomorphism \(f :
-G \to H\) is a homomorphism of Lie algebras, and the chain rule implies \((f
-\circ g)^* = f^* \circ g^*\). This is known as the \emph{the Lie functor}
+G \to H\) is a homomorphism of Lie algebras, and the chain rule implies \(d (f
+\circ g)_1 = d f_1 \circ d g_1\). This is known as the \emph{the Lie functor}
 \(\operatorname{Lie} : \mathbf{LieGrp} \to \mathbb{R}\text{-}\mathbf{LieAlg}\)
 between the category of Lie groups and smooth group homomorphisms and the
 category of Lie algebras.