Definition 60.4.1. Let $(A, I, \gamma )$ and $(B, J, \delta )$ be divided power rings. Let $A \to B$ be a ring map. We say $\delta $ is compatible with $\gamma $ if there exists a divided power structure $\bar\gamma $ on $J + IB$ such that
\[ (A, I, \gamma ) \to (B, J + IB, \bar\gamma )\quad \text{and}\quad (B, J, \delta ) \to (B, J + IB, \bar\gamma ) \]
are homomorphisms of divided power rings.
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