The fact that the category of divided power algebras has all coproduct is the (affine) building block for the fact that the crystaline site is actually a site. Given a diagram of pd-algebras $(A_i,I_i,\gamma_i)_{c\in \mathcal{C}}$, we study the functor $$\lim \text{Hom}((A_i,I_i,\gamma_i),(B,J,\delta))$$ and show that this is representable using the fact that the category of pd-algebras has all limits.
This coproduct however is, at the level of rings, NOT the tensor product of algebras (see https://stacks.math.columbia.edu/tag/07GY ). My question is whether or not there is a good way to see (explicitly) what the coproduct is?