Well-powered and co-well-powered categories, and supplied well-powerings

A subobject of an object $C$ is a mutual-factorisation class of monomorphisms into $C$ (Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms), and under the class convention of this development such a class is a formula rather than a set (Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{C}\mathbf{A}\mathbf{T}$ is not formed). So "the subobjects of $C$ form a set" cannot be stated by gathering the subobjects into a collection and measuring it. The size condition is stated on representatives instead, which is the form the sources use and the form every result below actually spends.

A category $\mathcal{C}$ is well-powered when, for every object $C$, there is a set $M_C$ of monomorphisms into $C$ (Monomorphism and epimorphism by left and right cancellation) containing a representative of every subobject class of $C$: every monomorphism into $C$ mutually factors with some member of $M_C$. It is co-well-powered when, for every $C$, there is a set of epimorphisms out of $C$ containing a representative of every quotient-object class.

A supplied well-powering gives such a set $M_C$ as data for every object at once — that is, the whole assignment $C\mapsto M_C$. A supplied co-well-powering is the dual datum. The difference from plain well-poweredness is not size but scope of selection: well-poweredness asserts of each object separately that a representative set exists, whereas a proof that needs one representative set per object across a proper class of objects would have to select them, and a supplied well-powering hands that assignment over rather than choosing it.