Corollary

Let E and F be two commutative monoids, S and T subsets of E and F respectively, f a homomorphism of E into F such that $f(S) \subset T$ and $\varepsilon : E \to E_S, \eta : F \to F_T$ the canonical homomorphisms. There exists one and only one homomorphism $g : E_S \to F_T$ such that $g \circ \varepsilon = \eta \circ f$.

The homomorphism $\eta \circ f$ of E into $F_T$ maps every element of S to an invertible element of $F_T$.