I want to show that if $f$ is an injective/surjective Lie group homomorphism than $f_*$ is injective/surjective.
$\require{AMScd}$ $$\begin{CD}G @>f>> H\\ @AexpAA @AAexpA\\ Lie(G) @>>f_*> Lie(H) \end{CD}$$
Take open sets $U,V,U', V'$ such that $V=exp(U), V'=exp(U'), f(V)=V', f_*(U)=U'$ and $exp|_U, exp|_V$ are diffeomorphisms.
injectivity : Suppose $f_*(x)=f_*(y)$ (where $f_*(x),f_*(y)\in V$. Then take $t$ small enough such that $tx,ty\in U$. Then by the diagram, we have $exp^{-1}(f(exp(tx)))=exp^{-1}(f(exp(ty)))$ so $tx=ty$ i.e. $x=y$.
surjectivity : Consider $y\in V\subset Lie(H)$. Now chase back in the diagram a preimage of $y$ with respect to $f_*$. Then since every $\hat{y}$ can be written as $ty$ for some $t\in \mathbb{R}$ and some $y \in V$ we are done.
Is this correct
What about the converse? f_* injective/surjective $\implies$ $f$ injective/surjective?