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It is forbidden to carry the Shofar on R”H for no need in any area that does not have an Eruv and is hence forbidden to carry on Shabbos. If however one originally carried the Shofar to Shul on Yom Tov then if he suspects the Shofar may get lost or stolen if he leaves it in Shul, he may return it home on Yom Tov, even if he has no more need to blow it that day. See Chapter 10 Halacha 25 for the full details of this subject!