Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have a morphism $\Phi:f^*[X,Y]\to[f^*X,f^*Y]$, though no extra conditions on this morphism ([SGA4, Tome 1, Exposé iv, 10.5][1]). Similarly, if $(\mathscr{E},A)$ is a ringed topos, $f^*$ does not preserve internal Ext, i.e. $$f^*\underline{\text{Ext}}^n_A(M,N)\ncong\underline{\text{Ext}}^n_{f^*A}(f^*M,f^*N),$$ where $\underline{\text{Ext}}^n_A(M,N)$ is the right derived functor of the [internal $A$-hom][2] $[M,N]_A$. Does there exist a correction to this, e.g. a spectral sequence that relates the two? 


  [1]: https://link.springer.com/book/10.1007/BFb0061319
  [2]: https://arxiv.org/pdf/2001.09075.pdf