This week we will discuss duality for equational/algebraic logic, the fragment of (many-sorted) first-order logic in which we only consider functional signatures and theories whose axioms are equations between terms. Previously we discussed the connection between algebraic theories and small categories with finite products, as well as the construction of the syntactic algebra (or "universal model”) of an algebraic theory. This week we will discuss the duality theorem for algebraic logic, namely the bi-equivalence between the 2-category of algebraic categories (categories equivalent to Mod(T) for some algebraic theory T) and the 2-category of “canonical” algebraic theories (algebraic theories that are idempotent-complete).