Borceux's Definition 4.1.2

Definition 4.1.2. A category A is well-powered when the subobjects of every object constitute a set.

contradicts Borceux's own Axiom 1.1.7 on page 3 of his book:

Axiom 1.1.7. A class is a set if and only if it belongs to some (other) class.

A large category (meaning the collection of objects is a proper class) can have a proper equivalence class of subobjects, which cannot be an element of any set. So a contractible groupoid with a proper class of objects is not a well-powered category in this definition, even though it is equivalent to the terminal category, which is well-powered.

The problem arises from the fact that the definition of a quotient of classes using equivalence classes is only correct when each equivalence class is a set. Otherwise one must define quotients using universal properties.

Thus, this problem can be resolved by using a categorical definition of a quotient instead of a set-theoretical one:

A category C is well-powered if for any object A∈C there is a surjective map Sub(A)→Q such that two subobjects of A are mapped to the same element of Q if and only if they are isomorphic, and, additionally, Q is a set (or a U-small set, etc.).

(Categorically, we could also say that Q is the quotient of Sub(A) with respect to the equivalence relation of isomorphism, where the quotient is defined using a universal property as the initial object in the category of maps Sub(A)→Q that send isomorphic subobjects to equal elements, without any reference to equivalence classes.)

Then Scott's trick, as explained in Andrej Bauer's answer, shows that other (correct) definitions are equivalent to this one.

Borceux's Definition 4.1.1

An equivalence class of monomorphisms with codomain A is called a subobject of A.

in combination with Definition 4.1.2

A category A is well-powered when the subobjects of every object constitute a set.

and the subsequent claim that the category of sets is well-powered contradicts Axiom 1.1.7 in his book:

A class is a set if and only if it belongs to some (other) class.

A large category (meaning the collection of objects is a proper class) can have a proper equivalence class of subobjects, which cannot be an element of any set. So a contractible groupoid with a proper class of objects is not a well-powered category in this definition, even though it is equivalent to the terminal category, which is well-powered.

The problem arises from the fact that the definition of a quotient of classes using equivalence classes is only correct when each equivalence class is a set. Otherwise one must define quotients using universal properties.

Thus, this problem can be resolved by using a categorical definition of a quotient instead of a set-theoretical one:

A category C is well-powered if for any object A∈C there is a surjective map Sub(A)→Q such that two subobjects of A are mapped to the same element of Q if and only if they are isomorphic, and, additionally, Q is a set (or a U-small set, etc.).

(Categorically, we could also say that Q is the quotient of Sub(A) with respect to the equivalence relation of isomorphism, where the quotient is defined using a universal property as the initial object in the category of maps Sub(A)→Q that send isomorphic subobjects to equal elements, without any reference to equivalence classes.)

Then Scott's trick, as explained in Andrej Bauer's answer, shows that other (correct) definitions are equivalent to this one.