First, We Consider Structures Freely Generated in Sem(Ξ)

A Ξ-structure F = (*F*, Φ<sup>F</sup> ) is freely generated in Sem(Ξ) by an Stm-sketch G = (*G*, *St*G) if, and only if, <sup>F</sup> is in Sem(Ξ) and there is a *valid interpretation* (*η*G, <sup>F</sup>) of <sup>G</sup> in <sup>F</sup> that is *universal relative to* Sem(Ξ). That is, for all Ξ-structures U = (*U*, Φ<sup>U</sup> ) in Sem(Ξ) and all valid interpretations (*ι*, U) of G in U there exists a unique morphism *ι* <sup>∗</sup> : F→U in Sem(Ξ) such that *η*G; *ι* ∗ = *ι* in Base, i.e., such that *ι* ∗ establishes an interpretation morphism *ι* <sup>∗</sup> : (*η*G, F) → (*ι*, U) in Int(G) Int(*G*) according to Definition 13.

A Ξ-structure, freely generated in Sem(Ξ) by an Stm-sketch G = (*G*, *St*G), is obviously uniquely determined "up to isomorphism in Sem(Ξ)" if it exists.

The universal property of (*η*G, F) entails that (*η*G, F) is initial in Int(G), thus the projection functor Π<sup>G</sup> : Int(G) → Sem(Ξ) establishes a functor from Int(G) into the co-slice category F/Sem(Ξ).

In the case that *St*<sup>G</sup> contains only atomic statements, the definition of morphisms between <sup>Ξ</sup>-structures ensures (*η*G; , <sup>U</sup>) <sup>|</sup>=*<sup>G</sup> St*<sup>G</sup> for any morphism : F→U in Sem(Ξ); thus, the assignments ( : F→U) → (*η*G; , U) establish a functor from F/Sem(Ξ) into Int(G). Due to the universal property of (*η*G, F), we obtain (*η*G; )<sup>∗</sup> = . Together with the equation *η*G; *ι* ∗ = *ι*, this ensures that the two functors establish an isomorphism between Int(G) and F/Sem(Ξ) (compare Proposition 4.10 in [2]). This justifies that we can call, in this *atomic case*, the pair (G, *η*G) a **sketch representation** of F.

Note that the Ξ-structure (*G*, Φ∅) with Φ<sup>∅</sup> a Φ-indexed family of empty sets is trivially freely generated in Sem(Ξ) by G = (*G*, ∅).
