Theorem: Neither the category $\mathcal{S}\mathrm{et}$ of sets nor the category $\mathcal{T}\mathrm{op}$ of topological spaces has a proper isomorphism-closed full subcategory that is both reflective and coreflective.
Theorem: Neither the category $\mathcal{S}\mathrm{et}$ of sets nor the category $\mathcal{T}\mathrm{op}$ of topological spaces has a proper isomorphism-closed full subcategory that is both reflective and coreflective.