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A class I of objects in a cartesian closed category C is called an exponential ideal if whenever Y∈I and X∈C, the exponential object Y X is in I.
Of course, in particular this implies that I is itself cartesian closed.
If I↪C is a reflective subcategory, then it is an exponential ideal if and only if its reflector C→I preserves finite products.
This appears for instance as (Johnstone, A4.3.1).
The relation of exponential ideals to reflective subcategories is discussed in section A4.3.1 of