Let $(M,+,e)$ be a commutative monoid with unit $e$. An element $a\in M$ is called cancellative element if

for any $b,c \in M$ such that $a+b=a+c$ implies that $b=c$.

Let $(\mathbf{N},+,0)$ the commutative monoid of natural numbers.

suppose that

1. we have two morphisms of monoids $f:(\mathbf{N},+,0)\rightarrow (M,+,e)$ and $g:(M,+,e)\rightarrow (\mathbf{N},+,0) $ such that $g\circ f= id$.
2. The monoid $(M,+,e)$ is torsion-free.

My question is the following: is the element $a=f(1)$ automatically a cancellative element in $(M,+,e)$ ?

Edit: By torsion-free I do mean that there does not exist a natural number $n>0$ and some element $x\in M-\{e\}$ such that $n x=e$.