An AB3 locally small abelian category with a generator is well-powered

Fix an object $A$. For each subobject $m:B\rightarrowtail A$, let $S_m\subseteq \mathcal{A}(G,A)$ be the subset of those maps $u:G\to A$ that factor through $m$. Because $\mathcal{A}$ is locally small, $\mathcal{A}(G,A)$ is a set, so its power set $\mathcal{P}(\mathcal{A}(G,A))$ is a set as well.

For each subset $S\subseteq \mathcal{A}(G,A)$, use AB3 and [L1] to form the canonical map $e_S:{\coprod }_{u\in S}G\to A$ and let $i_S:I_S\rightarrowtail A$ be its image. If $m:B\rightarrowtail A$ is any subobject, then [L1] applied to $B$ gives an epic canonical map ${\coprod }_{v\in \mathcal{A}(G,B)}G\twoheadrightarrow B\mathrm{.}$ Composing with $m$ produces exactly the family of maps in $S_m$, so the image of the resulting composite is $m$. Hence $i_{S_m}$ represents the same subobject as $m$.

The set of monomorphisms $\{i_S:I_S\rightarrowtail A\mid S\subseteq \mathcal{A}(G,A)\}$ therefore contains a representative of every subobject class of $A$. By [L2], $\mathcal{A}$ is well-powered.