Proof that the composition of two monoid homomorphisms is a monoid homomorphism.

Question

Let f : M → N and g : N → K be monoid homomorphisms. I tried to prove that g ∘ f : M → K is a monoid homomorphism too.

Let m,m' ∈ M then,

(g ∘ f) (mm') = g (f (mm'))

= g (f (m) f (m'))

= g (f (m)) g (f (m'))

= (g ∘ f) (m) (g ∘ f) (m')

Is that correct?