Take the category of presehaves of abelian groups on a topos $\mathcal{C}$. That is, an object of our category is a functor $F: \mathcal{C}^{\operatorname{op}} \to \operatorname{Ab}$.
We have a clear definition of a symmetric tensor product by $(F \otimes G)(S) = F(S) \otimes G(S)$.
However, defining an appropriate internal hom seems trickier. Taking inspiration from the internal sheaf hom, it seems like it would make sense to define
$${\mathscr{H}\kern -.5pt om} (F,G)(S) = \operatorname{Hom}(F_{| \mathcal{C}/S}, G_{|\mathcal{C}/S})$$
where $\mathcal{C}/S$ is the slice category. However, I can't seem to prove the adjunction (if it is even true). The main problem is that the slice category is more difficult to work with than in a general topos. Could there be some stronger requirements on the topos to make sure the adjunction is satisfied?
The category of abelian presheaves is locally finitely presentable, and for each abelian presheaf $F$, the functor $F \otimes {-}$ preserves colimits, so we may apply the accessible adjoint functor theorem to obtain a right adjoint $\mathscr{H}om (F, -)$. Representability of the functor $H \mapsto H (S)$ can be used to determine the values of $\mathscr{H}om (F, G)$: $$\mathscr{H}om (F, G) (S) \cong \textrm{Hom} (F \otimes \mathbb{Z} h_S, G)$$ Here $h_S = \mathcal{C} (-, S)$ and $\mathbb{Z} h_S$ is the free abelian presheaf generated by $h_S$.
Actually, your formula is also correct. The point is that you need to verify that $$\textrm{Hom} (F \otimes \mathbb{Z} h_S, G) \cong \textrm{Hom} (F |_{\mathcal{C}_{/ S}}, G |_{\mathcal{C}_{/ S}})$$ which can be done essentially the same way as the Yoneda lemma.