Lie group homomorphism determined by its differential ($\varphi (\exp X)=\exp (d_e\varphi (X))$)

A Lie group homomorphism intertwines one-parameter subgroups and therefore obeys the displayed identity. A connected Lie group is generated by an identity neighbourhood in an exponential chart, so its homomorphism is uniquely determined by the differential at the identity. On a disconnected group, only the restriction to its identity component is determined this way; the images of the other components require extra data.