Let E be a Grothendieck topos, such as the category of sheaves of sets on a topological space. Then there is a unique geometric morphism $(\Delta \dashv \Gamma)\colon E\to \mathrm{Set}$, where $\Delta\colon \mathrm{Set}\to E$ constructs constant sheaves and its right adjoint Γ takes global sections. If E is locally connected (i.e. Δ has a further left adjoint), then Δ is a cartesian closed functor, i.e. $\Delta(B^A)\cong \Delta B^{\Delta A}$ for sets A,B.

Now in any cartesian closed category, we can define the "object of isomorphisms" Iso(X,Y) between any objects X,Y, as an equalizer of a pair of maps $X^Y \times Y^X \rightrightarrows X^X \times Y^Y$. In particular, when X=Y, we have the object Aut(X) of automorphisms of X. Since inverse image functors preserve finite limits, if E is locally connected then $\Delta(\operatorname{Aut}(X))\cong \operatorname{Aut}(\Delta X)$ for any set X.

My question is twofold:

Is there a noticeably weaker condition on E than local connectedness which ensures that Δ preserves objects of automorphisms?

Can you give an explicit example of a topos for which Δ does not preserve objects of automorphisms?