Theorem 1.4.4 of Basic Lie Theory (Hossein Abbaspour Martin Moskowitz) states that for Lie groups $G$ and $H$ with $G$ simply conected every Lie algebra homomorphism $\phi: g \to h$ is the derivative of a Lie homomorphism $f:G \to H$ i.e $\phi = f'$.
The strategy of the proof uses a principle of Monodromy of Chevalley 1960s (Theorem $2$ and $3$ pag 47. Chevalley).
The problem I found is in the proof of Theorem 1.4.4 itself: given $\phi: g \to h$ and using the exponential map we can construct a local homomorphism $f$ between $G$ and $H$. In the following way, suppose that $U,V$ are the open respectively in $g$ and $h$ where $exp$ is local diffeomorphism such that $\phi(U) \subset V$. Then $g = exp X$ and we can define $f(g):= exp_H(\phi(X))$
I have no problem with the definition of the map but in order to prove that $f$ is an homomorphism (necessary in order to apply Theorem $2$ of Chevalley) it seems to me that we are requiring that $exp_G(X)\cdot exp_G(Y) = exp_G(X+Y)$ but I don't why this should be true in general.
Any help on this clarification would be appreciated.