A subobject of an object $X$ of a category $\mathsf{C}$ is an equivalence class of the equivalence relation $\equiv$ on the class of all monomorphisms with codomain $X$ where $f \equiv g$ whenever there is an isomorphism $h$ such that $g = f\circ h$.

 

A category $\mathsf{C}$ is well-powered if, for any $X \in \mathsf{C}$, all subobjects of $X$ form a set.

A subobject of an object $X$ of a category $\mathsf{C}$ is an equivalence class of the equivalence relation $\equiv$ on the class of all monomorphisms with codomain $X$ where $f \equiv g$ whenever there is an isomorphism $h$ such that $g = f\circ h$.

A category $\mathsf{C}$ is well-powered if, for any $X \in \mathsf{C}$, all subobjects of $X$ form a set.