Functor that does not preserve monic and epic In Jacobson’s book BAII, he gave two exercises to show that there exists functor that does not preserve monic or epic.

Ex-1.Let M and N be monoids as categories with a single object. Show that in this identification,a functor is a homomorphism of M into N.


Ex-2.Use Ex-1 to construct a functor $F$ and a monic(epic) $f$ such that $F(f)$ is not monic(epic)

I have solved Ex-1. However, I don’t know how to construct concrete example to solve Ex-2.
I do hope someone can give me some hints. Thank you very much!
Hint: Let $M$ be a free monoid.

 A simple example is with $M=(\Bbb N,+),\ \,N=(\{0,1\},\max)$ and $F(m)=\min(1,m)$.
 $1$ is cancellable in $M$ but not in $N$.
Hint: Find a monoid, and a homomorphism which takes an element with a left cancellative property to an element that doesn't.