In a topos, let $\mathsf{Ext}{\mathcal S}(X)\rightarrowtail\mathrm{Sub}(X) $ be defined as follows $$(U\rightarrowtail X)\in \mathsf{Ext}\mathcal S\iff \exists f:X\to R\text{ such that }U\cong f^{-1}R. $$
Let $\mathsf{Int}\mathcal S(X)\subset \Omega ^X$ be the internalizion of this, i.e defined by internally interpreting $$U\in \mathsf{Int}\mathcal S(X)\iff \exists f\in R^X\text{ such that }U= f^{-1}R. $$
Let $\Gamma$ denote global sections. Is it true that $\Gamma(\mathsf{Int}\mathcal S(X)\cong \mathsf{Ext}\mathcal S(X)$? If not, what is the relation between these subobjects? Are they isomorphic for representable $X$?