Definition
Category of complex Lie groups
Complex Lie groups and holomorphic group homomorphisms.
The category of complex Lie groups has complex Lie groups as objects and holomorphic group homomorphisms as morphisms. Its identity maps and composition are those of holomorphic maps. Holomorphicity is a requirement on the manifold map, not a claim that the group's carrier is a complex vector space.
Forgetting complex structure
Forgetting complex structure gives a faithful functor to real Lie groups. Entrywise conjugation on is a smooth real group automorphism but is antiholomorphic, so is not a morphism in this category. Its differential is real-linear and conjugate-linear on the complex Lie algebra.
Lie algebra
Differentiating at the identity produces a complex-linear bracket-preserving map. This distinguishes the complex Lie-algebra view from the larger real-linear vector-space endomorphism space.