*4.1. Representing 3/2-Institutions as Ordinary Institutions*

In [6], a general representation of stratified institutions as ordinary institutions was developed. In [14], it is shown that this constitutes a left adjoint functor from the category **SINS** of stratified institution morphisms to the category **INS** of ordinary institution morphisms. Let us recall this representation from either [6] or [14]. Given a stratified institution <sup>S</sup> = (*Sign*, *Sen*, *Mod*, [[\_]], <sup>|</sup>=), the following institution <sup>S</sup> = (*Sign*, *Sen*, *Mod* , <sup>|</sup><sup>=</sup> ) is defined by


$$\operatorname{Mod}^\sharp(\varrho)(\mathcal{M}',w') = (\operatorname{Mod}(\varrho)\mathcal{M}', [\![\mathcal{M}']\!]\_{\varrho}w');$$

• For each Σ-model *M*, each *w* ∈ [[*M*]]Σ, and each *ρ* ∈ *Sen*(Σ)

$$((M, w) \Vdash\_{\Sigma}^{\sharp} \rho) = (M \Vdash\_{\Sigma}^{w} \rho). \tag{13}$$

)

By "composing" the representation of 3/2-institutions as stratified institutions with the representation of stratified institutions as ordinary institutions, we obtain the following representation of 3/2-institutions as ordinary institutions.

**Corollary 2.** *Let* I = (*Sign*, *Sen*, *Mod*, |=) *be a 3/2-institution. Then,*

(Is ) = (*Sign*<sup>s</sup> , *Sen*<sup>s</sup> ,(*Mod*<sup>s</sup> ) , <sup>|</sup><sup>=</sup> )

*defines an ordinary institution where*

	- **–** *A* (*Mod*<sup>s</sup> ) *<sup>χ</sup>-model is pair* (*M*, *<sup>N</sup>*) *such that M* ∈ |*Mod*(*χ*✷)|*, N* <sup>∈</sup> *Mod*(*χ*)*M;*
	- **–** *A χ-model homomorphism* (*M*, *N*) → (*M*0, *N*0) *is a model homomorphism h* : *M* → *M*0*, such that N*<sup>0</sup> = *hN*✷*.*

$$(\mathcal{M}od^\circ)^\sharp(\varphi,\theta)(\mathcal{M}',\mathcal{N}') = (\mathcal{M}od(\theta)\mathcal{M}',\mathcal{M}od(\varphi)\mathcal{N}').$$

• *For each* (*Mod*<sup>s</sup> ) *χ-model* (*M*, *N*) *and each Sen*<sup>s</sup> *χ-sentence ρ*

$$(M, N) \equiv\_{\mathcal{X}}^{\sharp} \rho \quad \text{if and only if} \quad \mathbb{N} \doteq\_{\Box \times}^{\mathcal{T}} \rho.$$
