This is correct. The idea is that you can conjunctively (using the product) or disjunctively (using the coproduct) quantify the variables over the formulas that

This is correct. The idea is that you can conjunctively (using the product) or disjunctively (using the coproduct) quantify the variables over the formulas that use them, essentially getting a conjunction/disjuction of the formula, instantiated with each of the concrete values of the variable quantified over.