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