Consider a simply-connected Lie group $G$ with the maximal torus $H$ and Lie algebras $\mathfrak{g}$, $\mathfrak{h}$ respectively. The exponential map is $\exp(2\pi i \cdot) : \mathfrak{g} \to G$.
A cocharacter $X \in Hom(U(1), H)$ can be viewed as an element $X \in \mathfrak{h}$ such that $\exp(2\pi i X) = e \in G$. An element $\alpha^\vee$ in the coroot lattice $\Lambda_\text{coroot}$ can be equivalently viewed as an element $H^{\alpha^\vee} \in \mathfrak{h}$, which can be written as a linear combination of the Chevalley generators $h^i$ with integer coefficients: as a result, $\exp 2\pi i H^{\alpha^\vee} = e$, and therefore we have at least $\Lambda_\text{coroot} \subset \Lambda_\text{cocharacter}$.
I wonder, for simply connected $G$, do we actually have $\Lambda_\text{coroot} = \Lambda_\text{cocharacter}$? If so, how to see this?