Equalizer presentation by powers of a coseparator ($B⟶S^I\rightrightarrows S^J$)

In a complete category that is a locally small category with a coseparator $S$, suppose every monomorphism is a regular monomorphism. The evaluation embedding $B\to S^I$, $I=\mathcal{C}(B,S)$, is an equalizer of some pair into $Q$. Composing that pair with the evaluation embedding $Q\to S^J$ leaves the equalizer unchanged. Here powers mean products in a category indexed by sets, without assuming an exponential object exists.