  We prove a canonization result for the Carlson-Simpson forcing in the spirit
of \cite{KSZ}. We generalize the weak form of the Carlson-Simpson theorem
(\cite{CaSi}) dealing with partitions without free blocks: instead of dealing
with finite Borel (resp. Baire-property) colorings we deal with (uncountable)
colorings such that the corresponding equivalence relation (two partitions are
equivalent if they are colored by the same color) is analytic.
