Reading the book of Adamek and Rosicky, Locally Presentable and Accessible categories, I found an exercise I'm not able to solve (unless under some additional hypothesis).
Precisely, my problem is to prove the following statement:
Any (locally small) category with a strong generator is well-powered.
I recall my definitions: a set of objects $\mathcal G$ of a category $\mathcal C$ is said to be a strong generator if it is a generator and moreover for each object $K \in \text{Ob}(\mathcal C)$ and any proper subobject $m \colon K' \to K$ there is an element $G \in \mathcal G$ and a morphism $g \colon G \to K$ not factoring through $m$.
A category is said to be well-powered if for each object $K$, $\text{Sub}(K)$ is a set and not a proper class (I will assume to work in NBG).
Now, I'm able to prove this fact under the addtional hypothesis that $\mathcal C$ has pullbacks, but so far I haven't been able to figure out the general proof. I sketch my proof, just in the case it can be of some help in finding out the general one.
Fix an object $K \in \text{Ob}(\mathcal C)$ and consider $S := \coprod_{G \in \mathcal G} \text{Hom}_{\mathcal C}(G,K)$. This is a set because the category is locally small and $\mathcal G$ is a set by assumption. Define $F \colon \text{Sub}(K) \to \text{Parts}(S)$ by setting $F( m) := \{(G,f) \in S \mid f \text{ does not factor through }m\}$ I claim that $F$ is injective (and this will conclude). In fact, let $m_1 \colon K_1 \to K$ and $m_2 \colon K_2 \to K$ be two distinct subobjects. Let $m_3 \colon K_3 \to K$ be their intersection (i.e. the pullback of $K_1 \xrightarrow{m_1} K \xleftarrow{m_2} K_2$), let $j_i \colon m_3 \to m_i$ be the two canonical projections of the pullback. Then since $m_1$ is different (as subobject) from $m_2$, either $j_1$ or $j_2$ is not an isomorphism, let's say that $j_1$. Since $\mathcal G$ is a strong generator there is a morphism $g \colon G \to K_1$ not factoring through $j_1$ (for $G \in \mathcal G$); hence $m_1 \circ g$ cannot factor through $m_2$, otherwise the UMP of the pullback would force $g$ to factor through $j_1$, contradicting our construction. Hence $m_1 \circ g \in F(m_2)$ and $m_1 \circ g \not \in F(m_1)$, i.e. $F(m_2) \ne F(m_1)$.
How to get rid of the need for pullbacks?