Functor that does not preserve monic and epic

Question

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!

Answer 1

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 $N$.