First, We Consider Σ-Algebras Freely Generated in Alg(Σ)

A Σ-algebra F = (*F*, Ω<sup>F</sup> ) is freely generated by an Eq-sketch G = (*X*, *R*) if, and only if, there is a valid interpretation (*η*G, F) of G in F that is universal relative to Alg(Σ). That is, for all Σ-algebras A = (*A*, ΩA) and all valid interpretations (*ι*, A) of G in A there exists a unique morphism *ι* ◦ : F→A such that *η*G; *ι* ◦ = *ι* in Base*EQ*, i.e., such that *ι* ◦ establishes an interpretation morphism *ι* ◦ : (*η*G, F) → (*ι*, A) in Int(G) Int(*X*) (see Section 4.1).

$$\mathbf{Int}(\mathbb{G}) \qquad \qquad X \xrightarrow{(\eta \subset \mathcal{F}) \vdash\_{X} \mathbb{R}} \begin{vmatrix} \mathcal{X} \xrightarrow{(\eta \subset \mathcal{F}) \vdash\_{X} \mathbb{R}} F \\\\ \iota^{\circ} \\ A \end{vmatrix} \qquad \begin{vmatrix} \mathcal{F} = (F, \Omega^{\mathcal{F}}) & \mathcal{A} \lg(\Sigma) \\\\ \mathcal{A} = (A, \Omega^{\mathcal{A}}) \end{vmatrix}$$

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) → Alg(Σ) establishes a functor from Int(G) into the co-slice category F/Alg(Σ).

In contrast to Institutions of Statements, we have for arbitrary (!) Eq-sketches G = (*X*, *R*) (and not only for atomic Eq-sketches) that the definition of homomorphisms between Σ-algebras ensures (*η*G; , A) |=*<sup>X</sup> R* for any homomorphism : F→A in Alg(Σ) thus the assignments ( : F→A) → (*η*G; , A) establish a functor from F/Alg(Σ) into Int(G). Due to the universal property of (*η*G, F), we obtain (*η*G; )◦ = . Together with the equation *η*G; *ι* ◦ = *ι*, this ensures that the two functors establish an isomorphism between Int(G) and F/Alg(Σ) (compare Proposition 4.10 in [2]). This justifies that we can call the pair (G, *η*G) = ((*X*, *R*), *η*G) a sketch representation of F.

For arbitrary Eq-sketches G = (*X*, *R*), a Σ-algebra F, freely generated by G, exists and is uniquely determined "up to isomorphism". In the introductory Subsection 1.1.1, we used the notation F(Σ, ∅, *X*, *R*) to denote those freely generated Σ-algebras.

The Σ-algebra, freely generated by (*X*, ∅) is nothing but the Σ-term algebra TΣ(*X*) = F(Σ, ∅, *X*, ∅) on *X* and the unique morphism *ι* ◦ : TΣ(*X*) → A is simply the evaluation of terms ( see Equation (8)). In general, F(Σ, ∅, *X*, *R*) can be constructed as a quotient of TΣ(*X*).
