Every functor $\mathcal C\to\mathsf{Set}$ is an epimorphic image of a monofunctor implies every morphism of $\mathcal C$ is monic

Question

I am a trying to solve the following problem.

A functor $F: \mathcal{C} \rightarrow \mathsf{Set}$ is called a monofunctor if $F(f)$ is a monomorphism (that is, injective) for every morphism $f$ of $\mathcal{C}$.

Show that the following conditions on a small category $\mathcal{C}$ are equivalent:

1. Every morphism of $\mathcal{C}$ is monic.
2. Every representable functor $\mathcal{C} \rightarrow \mathsf{Set}$ is a monofunctor.
3. Every functor $\mathcal{C} \rightarrow \mathsf{Set}$ is an epimorphic image of a monofunctor.

Under what hypotheses on $\mathcal{C}$ is every functor $\mathcal{C} \rightarrow \mathsf{Set}$ a monofunctor?

I understand why $(1)$ and $(2)$ imply each other. For the implication $(2)\Rightarrow (3)$ I constructed an epimorphism for a presheaf $F$ $$\alpha:\coprod_{A\in \text{Ob }\mathcal{C}, \space x \in FA}\mathcal{C}(A,-)\twoheadrightarrow F $$

Since $\mathcal{C}(A,-)$ is a $\textit{monofunctor}$ for every $A\in\text{Ob }\mathcal{C}$, so is the domain of $\alpha$.

But I don't really know how to prove $(3)\Rightarrow (1)$. Can someone help me with this?

Answer 1

Hint 1

We have some $G:\mathcal C\to\mathbf{Set}$ such that $\alpha:G\twoheadrightarrow\mathcal{C}(A,-)$ for an object $A$. This means $\alpha_B:GB\twoheadrightarrow\mathcal C(A,B)$ for all $B$. In particular, for $\alpha_A$ we have a surjection $GA\twoheadrightarrow\mathcal C(A,A)$ which means there's an element $\eta\in GA$ such that $\alpha_A(\eta)=id_A$.

Hint 2

Naturality of $\alpha$ states $f\circ\alpha(x)=\alpha(Gf(x))$, and in particular $f=\alpha_B(Gf(\eta))$ for $f:A\to B$.

Answer

Next, let $g\circ h = g\circ k$ where $h,k:A\to B$. Clearly this means $$Gg\circ Gh=G(g\circ h)=G(g\circ k)=Gg\circ Gk$$ and so $Gg(Gh(\eta))=Gg(Gk(\eta))$. Since $Gg$ is injective because $G$ is a monofunctor, we have $Gh(\eta)=Gk(\eta)$. Therefore, $$h=\alpha_B(Gh(\eta))=\alpha_B(Gk(\eta))=k$$ so $g$ is a mono.