The Stacks project

Proof. The empty limit is the zero ring (that's weird but we need it). The product of a collection of divided power rings $(A_ t, I_ t, \gamma _ t)$, $t \in T$ is given by $(\prod A_ t, \prod I_ t, \gamma )$ where $\gamma _ n((x_ t)) = (\gamma _{t, n}(x_ t))$. The equalizer of $\alpha , \beta : (A, I, \gamma ) \to (B, J, \delta )$ is just $C = \{ a \in A \mid \alpha (a) = \beta (a)\} $ with ideal $C \cap I$ and induced divided powers. It follows that all limits exist, see Categories, Lemma 4.14.11. $\square$