Let $G$ and $H$ be Lie groups with identity elements $e_G$ and $e_H$, and let their Lie algebras be $\mathfrak g=T_{e_G}G$ and $\mathfrak h=T_{e_H}H$. Let $\varphi:G\to H$ be a smooth Lie [group homomorphism](/page/Group%20Homomorphism), and write $d\varphi:=d\varphi_{e_G}:\mathfrak g\to\mathfrak h$ for its differential at the identity. Then for every $X\in\mathfrak g$ and every $t\in\mathbb R$,

\begin{align*} \varphi(\exp_G(tX))=\exp_H(t\,d\varphi(X)). \end{align*}

In particular,

\begin{align*} \varphi(\exp_G X)=\exp_H(d\varphi(X)). \end{align*}