Second, We Consider Structures Freely Generated Relative to a Subcategory D:

Let D be an arbitrary full subcategory of Sem(Ξ). A Ξ-structure F = (*F*, Φ<sup>F</sup> ) is freely generated in <sup>D</sup> by an Stm-sketch <sup>G</sup> = (*G*, *St*G) if, and only if, <sup>F</sup> is an object in <sup>D</sup> and there is a valid interpretation (*η*G, F) of G in F that is universal relative to D. That is, for all Ξ-structures U = (*U*, Φ<sup>U</sup> ) in D and all valid interpretations (*ι*, U) of G in U there exists a unique morphism *ι* <sup>∗</sup> : F→U in D 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*).

A Ξ-structure, freely generated in D by an Stm-sketch G = (*G*, *St*G) is, obviously, uniquely determined "up to isomorphism in D" if it exists. In this case, the universal property of (*η*G, <sup>F</sup>) entails that (*η*G, <sup>F</sup>) is initial in the subcategory Int(G) <sup>↓</sup> <sup>D</sup> :<sup>=</sup> <sup>Π</sup>−<sup>1</sup> *<sup>G</sup>* (D) of all valid interpretations of G in Ξ-structures in D. Analogous to the case D = Sem(Ξ), we obtain, moreover, an isomorphism between Int(G) ↓ D and the co-slice category F/D if *St*<sup>G</sup> contains only atomic statements.

#### Third, We Consider Subcategories Described by Logical Means

One logical means to describe subcategories of Sem(Ξ) are Stm-sketches (**0**, *St*) with *St* only containing closed formulas, i.e., statements of the form (**0**, *Ex*, *id***0**). As discussed in Remark 21, those sketches can be seen as presentations in the traditional sense of the Theory of Institutions specifying subcategories Sem(Ξ,(**0**, *St*)) of Sem(Ξ).

As another logical means to describe subcategories of Sem(Ξ), we will introduce sketch implications in Section 5.2.3 (see Remark 27).
