Definition 4.14 (Subobject). By a subobject of $A\in \mathrm{ob}C$, we mean a monomorphism $A^{\prime }\rightarrowtail A$. We order subobjects by $(A^{\prime }\rightarrowtail A)\le (A^{″}\rightarrowtail A)$ if there exists

We write ${\mathrm{Sub}}_C(A)$ for this preorder.

We say $C$ is well-powered if every ${\mathrm{Sub}}_C(A)$ is equivalent to a small poset.