Monomorphism iff injective in the category of groups

Statement
The following are equivalent for a homomorphism of groups $$\alpha: H \to G$$:


 * 1) $$\alpha$$ is injective as a set map.
 * 2) $$\alpha$$ is a monomorphism with respect to the category of groups: For any homomorphisms $$\theta_1,\theta_2:K \to H$$ from any group $$K$$, $$\alpha \circ \theta_1 = \alpha\ circ \theta_2 \implies \theta_1 = \theta_2$$.

Related facts

 * Epimorphism iff surjective in the category of groups

Injective homomorphism implies monomorphism
This follows simply by thinking of the maps as set maps. In general, for any concrete category, any injective homomorphism is a monomorphism.

Monomorphism implies injective homomorphism
Given: $$\alpha$$ is a monomorphism: For any homomorphisms $$\theta_1,\theta_2:K \to H$$ from any group $$K$$, $$\alpha \circ \theta_1 = \alpha\ circ \theta_2 \implies \theta_1 = \theta_2$$.

To prove: $$\alpha$$ is injective, i.e., the kernel of $$\alpha$$ is the trivial subgroup of $$G$$.

Proof: