The first isomorphism theorem implies that any surjective homomorphism ϕ ϕ : G → → H factors canonically as a quotient homomorphism followed by an isomorphism: ⁠ G → → G / ker ⁡ ⁡ ϕ ϕ → → ∼ ∼ H ⁠. Surjective homomorphisms are the epimorphisms in the category of groups.