= Products and coproducts of fixed cardinality in a topos
{title2=$\kappa\text{-products}\iff\kappa\text{-coproducts}$}
For an <elementary topos>, existence of <products in a category> indexed by a fixed set is equivalent to existence of the corresponding <coproducts in a category>. The <power-object monadicity> equivalence $\mathcal E^{\mathrm{op}}\simeq\mathcal E^{PP}$ gives the forward direction because <Eilenberg-Moore categories> create limits. For the converse, the diagonal into the pointwise product category is a <logical functor>. A coproduct is its <left adjoint>; the <adjoint lifting theorem for monad algebra functors>, applied through power-object monadicity, then gives its <right adjoint>, namely the product. The required reflexive coequalizers on the algebra side are opposites of finite equalizers in the topos.
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