Theorem: The forgetful functor $\mathcal{T}\mathrm{op}\to\mathcal{S}\mathrm{et}$, $\mathcal{G}\mathrm{rp}\to\mathcal{S}\mathrm{et}$, $\mathcal{R}\mathrm{ing}\to\mathcal{A}\mathrm{b}$, $\mathcal{T}\mathrm{op}*\to\mathcal{T}\mathrm{op}$ are faithful but not full.\nomenclature{$\mathcal{S}\mathrm{et}$}{the category of sets}\nomenclature{$\mathcal{A}\mathrm{b}$}{the category of abelian groups}\nomenclature{$\mathcal{T}\mathrm{op}*$}{the category of topological spaces with a based point}\nomenclature{$\mathcal{T}\mathrm{op}$}{the category of topological spaces}