 Exponentials in a coalgebra topos

For a finite-limit-preserving comonad on a topos, start with the cofree coalgebra on the ambient exponential $B^A$. Its underlying evaluation is $e=\mathrm{ev}(ϵ\times 1)$. Transpose the two expressions $be$ and $Ge(\delta \times a)$ first using the ambient exponential and then the cofree adjunction. Their coalgebra equalizer represents exactly the maps whose evaluations respect coalgebra structure. Thus coalgebra exponentials need not be the underlying ambient exponentials.