Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of the fiber of $f$ over $t$, and let $\mathcal{F}$ a coherent sheaf on $X$ that is flat over $C$.

Does there exist an isomorphism $i_t^{*}\mathcal{E}xt^1(\mathcal{F},\mathcal{O}_X)\cong \mathcal{E}xt^1(i_t^{*}\mathcal{F},\mathcal{O}_{X_t})$ ?