Reading a book about curves I encoutered the following claim, which I don't understand.
Let $X$ be a smooth projective curve, and $\nu:X\times \mathrm{Pic}(X)\to \mathrm{Pic}(X)$. Pick a universal Poincare' line bundle $\mathscr{L}$ and a divisor $D$ of high degree on $X$, and let $\Gamma = D\times \mathrm{Pic}(X)$. Consider the short exact sequence $$ 0 \to \mathscr{L} \to \mathscr{L}(\Gamma) \to \mathscr{L}(\Gamma)/\mathscr{L} \to 0 $$ and take the direct image to $\mathrm{Pic}(X)$ via $\nu$ to get $$ 0 \to \nu_*\mathscr{L} \to \nu_*\mathscr{L}(\Gamma) \to \nu_*\mathscr{L}(\Gamma)/\mathscr{L} \to R^1\nu_*\mathscr{L} \to 0. $$
Act similarly for a family of line bundles $L$ parametrised by a scheme $T$, letting $\phi:X\times T \to T$. We get
$$ 0 \to \phi_*L \to \phi_*L(\Gamma') \to \phi_*L(\Gamma')/L \to R^1\phi_* L \to 0. $$
By the universal property of $\mathscr{L}$ we have a unique map $f:T\to \mathrm{Pic}(X)$ such that $f^*\mathscr{L} = L\otimes \phi^*E$ for a line bundle $E$ on $T$.
The author of my book claims the following:
applying the proper base change theorem to the map $\nu$ and the sheaf $\mathscr{L}(\Gamma)$, it follows that the natural map $$ f^*\nu_* \mathscr{L}(\Gamma) \to \phi_*(id\times f)^*\mathscr{L}(\Gamma) $$ is in fact an isomorphism.
Could you help me seeing how the theorem is applied, in this specific situation?