Is the dual of a flat module flat

Question

Let $k$ be an algebraically closed field, $T$ be a integral, regular, projective $k$-scheme and $X$ another projective, integral $k$-scheme. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k T$, which is flat over $T$. Is the dual sheaf $\mathcal{Hom}_{\mathcal{O}_{X \times_k T}}(\mathcal{F},\mathcal{O}_{X \times_k T})$ going to be flat over $T$?