= Power of sets as a capital regular category
{title2=$\mathbf{Set}^{I}$}
Here $I$ is a <set> regarded as a discrete <category>. <Finite limits> and <regular epimorphisms> are coordinatewise. A <well-supported object> is a family of nonempty <sets>. With the <axiom of choice>, every coordinate element extends to a global point of its product. Any <subobject> containing all global points must therefore contain every coordinate element, and be the whole family. Thus the category is capital. Families with some empty coordinates explain why global points need not reflect arbitrary <isomorphisms>.
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