= Conservative regular representation in sets
{title2=$V:\mathcal C\to\mathbf{Set}$}
For a <regular category> that is a <small category>, a <regular functor> to the <Category of sets> that reflects <isomorphisms> exists exactly when the category has <almost total support for a regular category>. Necessity follows because a nonempty image forces the object's support to be terminal, while an empty image makes all projections $A\times X\to A$ invertible and makes all equalizers of maps out of $A$ invertible. This proves that $A$ is strict initial. For sufficiency, use the capitalization theorem to map conservatively and regularly into a capital regular category, then take <global sections of a capital regular category>. Well-supported objects retain points; proper strict initial objects retain empty point sets. A bijection between nonempty point sets makes the image subobject entire and makes the kernel-pair diagonal entire, by well-pointedness. Thus the image arrow is both a regular epimorphism and a monomorphism, and is invertible. The capitalization theorem is a substantive existence input; this argument does not infer arbitrary large-category representability from small-category hypotheses.
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