Let $\varphi$ be an $\tau$-sentence, we define the generalized spectrum of $\varphi$ as the class of its finite models, $$\text{GenSpec}(\varphi):=\{\mathcal{A}; \mathcal{A} \models \varphi, \lvert A\rvert < \aleph_0\}$$ and the spectrum of $\varphi$ as the set of cardinalities of finite models $$\text{Spec}(\varphi):=\{\lvert A \rvert; \mathcal{A}\in\text{GenSpec}(\varphi)\}.$$

It is well known that coding $\text{Spec}(\varphi)$ using binary strings results in precisely all $\text{NE}$ sets, this follows from the Fagin's theorem which states that the set of binary encoded linearly ordered structures of $\text{GenSpec}(\varphi)$ are precisely the $\text{NP}$ sets of binary encoded strctures.

Equip $\text{GenSpec}(\varphi)$ with all $\tau$-structure morphisms and call this category $\text{MOD}_{\text{fin}}(\varphi)$.

**The question:** Is it known whether for every $\varphi$ an $\tau$-sentence there exists $\psi$ a $\tau$-sentence such that $$\text{Spec}( \varphi)=\text{Spec}(\psi),$$
and $\text{MOD}_{\text{fin}}(\psi)$ is a grupoid? In other words, do morphisms matter while considering spectra of FO sentences?