Why the category of core-compact spaces with continuous maps is not cartesian closed ?

According to ESCARDÓ-LAWSON-SIMPSON paper 'Comparing cartesian closed categories of (core) compactly generated spaces' The following four propositions are true:

• A topological space $X$ is exponentiable iff $X$ is core-compact.

• The category of core-compact topological spaces is not cartesian closed, because even though when $X$ and $Y$ are core-compact, we can build the exponential object $Y^X$, but $Y^X$ will not necessarily be itself core-compact.

• The category of Core-compactly generated spaces is cartesian closed.

• The category of Core-compactly generated spaces contains the category of core-compact spaces.

It seems that there is a contradiction if the four propositions are all true. A sub-category of the category of topological spaces cannot be cartesian closed if it is not strictly included in the sub-category of core-compact spaces. What Am i getting wrong here ?

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The way I read this is that an exponential $Y^X$ of two core compact spaces may fail to be core-compactly generated. –  Andrej Bauer Dec 5 '11 at 14:10
"Exponentiable" is relative to the ambient category, and a non-exponentiable object in a category $C$ may turn out to be exponentiable in some subcategory. –  Qiaochu Yuan Dec 5 '11 at 14:11