The Stacks project

Lemma 23.3.3. Let $\mathcal{C}$ be the category of divided power rings. Let $F : \mathcal{C} \to \textit{Sets}$ be a functor. Assume that

  1. there exists a cardinal $\kappa $ such that for every $f \in F(A, I, \gamma )$ there exists a morphism $(A', I', \gamma ') \to (A, I, \gamma )$ of $\mathcal{C}$ such that $f$ is the image of $f' \in F(A', I', \gamma ')$ and $|A'| \leq \kappa $, and

  2. $F$ commutes with limits.

Then $F$ is representable, i.e., there exists an object $(B, J, \delta )$ of $\mathcal{C}$ such that

\[ F(A, I, \gamma ) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {C}((B, J, \delta ), (A, I, \gamma )) \]

functorially in $(A, I, \gamma )$.

Proof. This is a special case of Categories, Lemma 4.25.1. $\square$

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