Is there a geometric theory $T$ and a Grothendieck topos $\mathcal E$ such that (2) holds but (1) doesn't:

$\mathcal E$ 2-represents the 2-functor $$\mathbf{GrothTop}\to\mathbf{Cat}$$ which sends a Grothendieck topos $\mathcal E$ to the category of models of $T$ in $\mathcal E$.

$\mathcal E$ represents the 1-functor $$\mathrm h\mathbf{GrothTop}\to\mathrm h\mathbf{Cat}\to \mathbf{Set}$$ which can be obtained by truncating the above functor to the 1-categorical level and composing with the functor sending a category $\mathcal C$ to the set $\mathrm{Ob}(\mathcal C)/\mathord{\cong}$ of all objects of $\mathcal C$ up to isomorphism.

Conversely, (1) implies (2), right?