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Feb 10, 2019 at 23:43 comment added Mike Shulman Unfortunately I can only accept one answer, but this one adds some extra to Karol's: you can choose an even simpler cartesian monoidal $P = (\cdot \to\cdot)$ than his, and rather than acting on a small $J$ and extending to presheaves by convolution, you can have it act directly on a locally presentable category by cocontinuous functors.
Feb 10, 2019 at 6:29 history answered Reid Barton CC BY-SA 4.0