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Do you think it should be as follows?

Consider the set of maps: Map(X, Y), where f belongs to Map if and only if f maps X to Y.
All the partitions are the union of preimages(corresponding to the f mentioned later) of y, where y belongs to the images of all possible f's.

All I'm trying to say is that ALL maps such that X -> Y should be considered.
What you wrote, in my opinion, is just some case of partitions, which will not be all of them, since it is derived from ONE possible map from X to Y.

Thanks,

Xi Chen