= Capitalization of a small regular category
{title2=$\mathcal C\longrightarrow\mathbf{Set}^{I}$}
Choose one <regular functor> witnessing the <regular-logic separation of a proper subobject> for each proper <subobject> represented by a <monomorphism> of the small <regular category>. Their product is regular and reflects invertibility of monomorphisms. If an arbitrary <morphism> becomes invertible, its image inclusion becomes invertible and so does the diagonal of its <kernel pair>. The original <morphism> is both a <regular epimorphism> and a <monomorphism>, hence invertible. The target is a <power of sets as a capital regular category>. This proves an isomorphism-reflecting regular representation without asserting full faithfulness.
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