Proposition 99.4.3. In Situation 99.3.1 assume that
$f$ is of finite presentation, and
$\mathcal{F}$ and $\mathcal{G}$ are finitely presented $\mathcal{O}_ X$-modules, flat over $B$, with support proper over $B$.
Then the functor $\mathit{Isom}(\mathcal{F}, \mathcal{G})$ is an algebraic space affine of finite presentation over $B$.
Proof. We will use the abbreviations $H = \mathit{Hom}(\mathcal{F}, \mathcal{G})$, $I = \mathit{Hom}(\mathcal{F}, \mathcal{F})$, $H' = \mathit{Hom}(\mathcal{G}, \mathcal{F})$, and $I' = \mathit{Hom}(\mathcal{G}, \mathcal{G})$. By Proposition 99.3.10 the functors $H$, $I$, $H'$, $I'$ are algebraic spaces and the morphisms $H \to B$, $I \to B$, $H' \to B$, and $I' \to B$ are affine and of finite presentation. The composition of maps gives a morphism
of algebraic spaces over $B$. Since $I \times _ B I' \to B$ is separated, the section $\sigma : B \to I \times _ B I'$ corresponding to $(\text{id}_\mathcal {F}, \text{id}_\mathcal {G})$ is a closed immersion (Morphisms of Spaces, Lemma 67.4.7). Moreover, $\sigma $ is of finite presentation (Morphisms of Spaces, Lemma 67.28.9). Hence
is an algebraic space affine of finite presentation over $B$ as well. Some details omitted. $\square$
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