I have been working through a multi-sorted, categorical presentation of equational logic. Given a collection of equations in context $ Th $, there is a syntactic construction of the "classifying category" denoted $ Cl(Th) $ which is a category with finite products, that has a canonical Th-algebra, $ G $ called the "generic $Th$-algebra" with the following property: Each $Th$-algebra $ S $ in another category $ \mathcal{C} $ with finite products is the "image" of $ G $ under a finite product preserving functor $ \overline{S} \colon Cl(Th) \to \mathcal{C} $. Furthermore, one can show that this establishes a categorical equivalence between the category of $ Th $-algebras and homomorphisms in $\mathcal{C}$ and the category of finite product preserving functors from $ Cl(Th) $ to $ \mathcal{C}$ and natural transformations.
I can work through the details of the above. However, it is mentioned that the classifying category and generic $ Th $-algebra have universal properties which characterize them up to equivalence and up to isomorphism respectively. I think I understand what the universal properties should be but I am having trouble nailing down a proof that it characterizes the $Cl(Th)$ up to equivalence and $ G $ up to isomorphism. The following link Andrew Pitts Categorical Logic is to the notes I am working through.
Thank you for your consideration.