Example of a functor that doesn't reflect isomorphism

Question

I'm reading the text Category theory in context from Emily Riehl, and having trouble to find an example asked on exercise $1.3.viii.$: prove that functors not need to reflect isomorphisms, i.e., find a functor $F:C\rightarrow D$ and a morphism $f$ in $C$ so that $Ff$ is an isomorphism in $D$ but is not an isomorphism in $C$.

I know that a non conservative functor from $Top$ to $Set$ might work but can't find the adequate morphism.

Any suggestion is apreciated.

Answer 1

A very generic counterexample:

Take for $\mathcal{D}$ the category with one object $O$ and one morphism $f$. There is a functor $F$ from any category to $\mathcal{D}$ that sends every object to $O$ and any morphism to $f$.

Since $f$ is an isomorphism this generates a counterexample for every category which contains at least one morphism which is not an isomorphism.

Answer 2

Hint: look at any continuous bijection that fails to be a homeomorphism.

Answer 3

Just take the forgetful functor $F$ from $\mathit{Top}$ to $\mathit{Set}$. Then, take a bijection $f$ between two topological spaces which is not a homeomorphism. Of course, $Ff$ will be an isomorphism.

Answer 4

Homology and homotopy functors are very natural examples of this.