Definition of "homomorphism"

This motivates a generalization, and exponential homomorphisms are now defined, in an algebraic fashion, from certain free products to formal power series rings with non-commutative indeterminates.

1954, Kuo-Tsai Chen, Iterated Integrals and Exponential Homomorphisms, Proceedings of the London Mathematical Society, Reprinted in 2001, Philippe Tondeur (editor), Collected Papers of K.-T. Chen, Birkhäuser, page 54

A homomorphism of presheaves h : A → B {\displaystyle h:A\rightarrow B} is a collection of homomorphisms h U : A ( U ) → B ( U ) {\displaystyle h_{U}:A(U)\rightarrow B(U)} commuting with restrictions.

1997, Glen E. Bredon, Sheaf Theory, 2nd edition, Springer, page 8

Definition 1.15. Let G {\displaystyle G} and H {\displaystyle H} be matrix Lie groups. A map Φ {\displaystyle \Phi } from G {\displaystyle G} to H {\displaystyle H} is called a Lie group homomorphism if (1) Φ {\displaystyle \Phi } is a group homomorphism and (2) Φ {\displaystyle \Phi } is continuous.

2003, Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, page 17