Well-powered category

Remarks

Example: the category of sets is well-powered. Each subobject of a set $X$ is represented by the inclusion of a subset. So the subobjects of $X$ are in bijection with the power set of $X$, which is a set. The same holds for groups, rings and modules, where subobjects correspond to substructures. In the category of topological spaces, a subobject of $X$ corresponds to a subset of $X$ with a topology finer than the subspace topology. These pairs also form a set.

Non-example: let $\mathcal{C}$ have as objects all sets and one more object $X$. Besides identities, let $\mathcal{C}$ have exactly one morphism $m_S \colon S \to X$ for each set $S$, and nothing else. It is a category, since no two non-identity morphisms compose. Each $m_S$ is a monomorphism, since there are no two different morphisms into $S$ from one object, and $m_S \equiv m_T$ only when $S = T$, since there is no morphism between distinct sets. So the subobjects of $X$ form a proper class, and $\mathcal{C}$ is not well-powered.