A locally small category that is not well-powered: one object admits no set of representative monomorphisms

Put one morphism $x\to y$ exactly when $x\le y$ in the displayed order. Every hom-collection is therefore empty or a singleton, so the definable-class category is locally small by [L5].

Every morphism in a thin category is monic: any parallel arrows that can be composed with it are already equal. Hence each arrow $m_{\alpha }:\alpha \to \mathrm{\infty }$ represents a subobject of $\mathrm{\infty }$.

The arrows $m_{\alpha }$ and $m_{\beta }$ mutually factor exactly when both $\alpha \le \beta$ and $\beta \le \alpha$, hence exactly when $\alpha =\beta$. Thus distinct ordinals give distinct subobject classes.

Suppose some set $M$ of monomorphisms into $\mathrm{\infty }$ contained a representative of every subobject class. By step 2.2 the only monomorphism into $\mathrm{\infty }$ that mutually factors with $m_{\alpha }$ is $m_{\alpha }$ itself, so $M$ would have to contain $m_{\alpha }$ for every ordinal $\alpha$, and $\alpha \mapsto m_{\alpha }$ is injective. Sending each such member of $M$ back to its domain would then exhibit the ordinals as the image of a set, making them a set and contradicting [L3]. No such $M$ exists, so the category is not well-powered by [L4], despite being locally small.