Naturality of the Lie group exponential ($F({\exp }_{G}v)={\exp }_H(D_{e}Fv)$)

For a Lie group homomorphism $F$, the image of the integral curve through the identity of a left-invariant vector field is the integral curve of the field induced by its differential. Uniqueness of integral curves of a vector field identifies the curves for all times. Evaluating at time one gives the formula. Completeness of invariant fields follows from left translation and repetition of a local integral curve, independently of a matrix-series formula.