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Valery Isaev (Jan 27 2025 at 13:36):After a recent conversation here about functorial definition of schemes, I've got interested in small presheaves. It turns out it is a non-trivial question whether a category of small presheaves has some nice categorical properties. For example, Rosický in this paper gives a criterion for this category to be Cartesian closed.
Now, I wonder whether the category of small presheaves on has some nice categorical properties. Is it Cartesian closed? Is it a pretopos? There is a related Mathoverflow question, but it mainly discusses when the category of small presheaves is a topos.