Implication details

Proof: For an elementary topos $\T$, the power object functor $P : \T^{\op} \to \T$ is right adjoint to $P^{\op} : \T \to \T^{\op}$ and moreover is faithful. The first statement is part of the proof that $P$ is monadic (Johnstone, 2.2.7) and the second is a corollary. Therefore, from Lemma 9 here, if $F$ is a generating collection of $\T$, then $P_*(F)$ is a cogenerating collection.