the functor ${\operatorname{Hom}}_{R'}\left(-,N\otimes_{R}R'\right)$ the thesis follows as $${\operatorname{Hom}}_{R'}\left(F_{\bullet}\otimes_{R}R',N\otimes_{R}R'\right)\simeq{\operatorname{Hom}}_{R}\left(F_{\bullet},N\right)\otimes_{R}R'\,.$$

We note that, if ${\mathcal{E}}$ is flat on $S$, for every point $s\in S$ there are isomorphisms $$\left[{\operatorname{\mathcal{E}\textit{xt}}}\,\!^i_{{\mathcal{O}}_T}\left({\mathcal{E}},{\mathcal{O}}_T\right)\right]_s \simeq{\operatorname{\mathcal{E}\textit{xt}}}\,\!^i_{{\mathcal{O}}_{T_s}}\left({\mathcal{E}}_s,{\mathcal{O}}_{T_s}\right)\qquad i\geq0\,.
\label{eq1''}$$ In particular, for all $s\in S$ one has $\left({\mathcal{E}}^*\right)_s\simeq\left({\mathcal{E}}_s\right)^*$. Since $\dim X=2$ and $X$ is smooth it follows that $\left({\mathcal{E}}^*\right)_s$ is locally free on $T_s$.

The dual ${\mathcal{E}}^*$ of a coherent ${\mathcal{O}}_T$-module ${\mathcal{E}}$ flat on $S$ is locally free.

We claim that, for any point (closed or not) $s\in S$ one has $({\mathcal{E}}^*)_s\simeq({\mathcal{E}}_s)^*$. Indeed one has $$\begin{aligned}
({\mathcal{E}}^*)_s&={\operatorname{\mathcal{H}\!\textit{om}}}_{{\mathcal{O}}_T}({\mathcal{E}},{\mathcal{O}}_T)\otimes_{{\mathcal{O}}_T}{\mathcal{O}}_{T_s}\\
({\mathcal{E}}_s)^*&={\operatorname{\mathcal{H}\!\textit{om}}}_{{\mathcal{O}}_{T_s}}\left({\mathcal{E}}\otimes_{{\mathcal{O}}_T}{\mathcal{O}}_{T_s},{\mathcal{O}}_{T_s}\right)
\end{aligned}$$ Since ${\mathcal{O}}_{T_s}={\mathcal{O}}_T\otimes_{{\mathcal{O}}_S}k(s)$ and ${\mathcal{E}}$ is flat on $S$, the claim follows from Lemma \[pro1\].

In particular, whenever $s$ is *closed*, ${\mathcal{E}}_s^*$ is locally free since $\dim X=2$. This implies that, for all closed points $x\in T_s$, the stalk $({\mathcal{E}}_s)^*_x$ is a free ${\mathcal{O}}_{T_s,x}$-module. By equation it follows that for all closed points $(x,s)\in T$ the stalk ${\mathcal{E}}^*_{(x,s)}$ is a free ${\mathcal{O}}_{T_{(s,x)}}$-module. By Exercise 6.2 in [@Eis], ${\mathcal{E}}^*$ is locally free.

\[torsionfree\] Let ${\mathcal{E}}$ be a coherent ${\mathcal{O}}_T$-module such that the restrictions ${\mathcal{E}}_s$ are torsion-free for all closed $s\in S$. Then ${\mathcal{E}}$ is torsion-free.

By hypothesis, for all $x\in T_s$, the stalk $({\mathcal{E}}_s)_x$ is torsion-free. By equation , for all closed $(x,s)\in 