Homomorphism of algebraic groups

Concrete definition
Suppose $$G_1$$ and $$G_2$$ are algebraic groups over a field $$k$$. A 'homomorphism of algebraic groups is a function $$\varphi:G_1 \to G_2$$ satisfying both these conditions:


 * 1) $$\varphi$$ is a defining ingredient::homomorphism of groups.
 * 2) $$\varphi$$ is a morphism of algebraic varieties.