 Exponentiability test using a coseparator

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.