Compact open topology

Hello
it is true that there is a natural (in spaces X and Y) homeomorphism
hom(X x Y, Z) ---> hom(Y, hom(Y,Z))
since X, Y Hausdorff and X locally compact and the hom sets are
equipped with the compact open topology.
This is sometimes called the "exponential law".
I have some questions concerning this:
1. Does anybody know an easy counterexample of the above being a
homeomorphism in general?
2. I have read that the above map is always continuous and the
property of being bijective arises from the locally compactness of X.
I can't believe that this is true, because I think that the above map
is always bijective. Is this true?
3. Is the above map natural in Z, too?

Thanks
Sancho
.

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