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Suppose $ X $ and $ T $ are complex algebraic varieties and let $ Y $ be a subvariety of $ X $ . If we have a coherent sheaf $\mathcal {F} $ on the product $ X\times T $ flat over $ X $, is it true that the restriction of $\mathcal {F} $ to $ Y\times T $ is also flat?

Suppose $ X $ and $ T $ are complex algebraic varieties and let $ Y $ be a subvariety of $ X $ . If we have a coherent sheaf $\mathcal {F} $ on the product $ X\times T $ flat over $ X $, is it true that the restriction of $\mathcal {F} $ to $ Y\times T $ is also flat over $ Y $?