A class of objects in a cartesian closed category is called an exponential ideal if whenever and , the exponential object is in .
If is a reflective subcategory, then it is an exponential ideal if and only if its reflector preserves finite products.
This appears for instance as (Johnstone, A4.3.1). See also at reflective subuniverse. Note that in this case is itself a cartesian closed category, since being a reflective subcategory it is also closed under finite products.
The relation of exponential ideals to reflective subcategories is discussed in section A4.3.1 of
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