Compatibility Casefold Equivalence

Hi,

In Chapter 3 Section 13, the Unicode spec defines D146:

"A string X is a compatibility caseless match for a string Y if and only if: NFKD(toCasefold(NFKD(toCasefold(NFD(X))))) = NFKD(toCasefold(NFKD(toCasefold(NFD(Y)))))"

I am trying to understand the "if and only if" part of this.   Specifically, why is the outermost NFKD necessary?  Could it also be a NFKC normalization?   Is wrapping the outer NFKD in a NFC or NFKC on both sides of the equation okay?

My use case is that I am trying to store user-provided tags in a database.  I would like the tags to be deduplicated based on compatibility and caseless equivalence, which is how I ended up looking at D146.  However, because decomposition can result in much larger strings, I would prefer to keep  the stored version in NFC or NFKC (I *think* this doesn't matter after doing the casefolding as described above).

Thanks,

Carl