= Global sections of a capital regular category
{title2=$\Gamma=\mathcal C(1,-)$}
In a <locally small> <capital regular category>, $\Gamma$ is a <regular functor> and reflects <terminal objects>. A well-supported object $A$ has a point: otherwise $A\times A$ is well-supported with no points, so its diagonal is invertible by well-pointedness; then $A$ is subterminal and its regular epimorphism to $1$ is invertible, a contradiction. Pulling a regular epimorphism back along a point gives a well-supported object, so the point lifts. This proves preservation of regular epimorphisms, while representability gives limit preservation. If $\Gamma X$ is a singleton, its unique point $1\to X$ is monic and contains all points; $X$ is well-supported, so that point is invertible. Global sections need not reflect arbitrary isomorphisms between objects of proper support.
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