= Exponentiability test using a coseparator
{title2=$\mathcal C(-\times A,S)$ representable}
In a <complete category> that is a <locally small category> with a <coseparator> $S$ and all <monomorphisms> regular, an object $A$ is an <exponentiable object> exactly when $\mathcal C(-\times A,S)$ is a <representable functor>. A representing object $E$ gives representations $E^I$ for maps into powers $S^I$. An <equalizer presentation by powers of a coseparator> then constructs a representation for maps into every target. The <Yoneda lemma> makes these representations functorial and supplies the <right adjoint>.
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