A category \({\bf Fol}\) of sets of formulas of first-order languages 
with finitary relations and with equality is constructed. 
An adjunction \(\lan F,U,\eta,\epsilon\ran:{\bf HSet}\ra {\bf Fol},\) 
where \({\bf HSet}\) denotes the category of families of sets indexed 
by subsets of the natural numbers``respecting inclusions" is then obtained. 
It gives rise to an algebraic theory \({\bf T}\) over \({\bf HSet}.\) 
It is then shown that the Eilenberg-Moore category of \({\bf T}\)-algebras 
in\({\bf HSet}\) has a subcategory isomorphic to the category
\(\overrightarrow{{\rm Lf}_{\omega}}\) corresponding to the
variety \({\rm Lf}_{\omega}\) of all \(\omega\)-dimensional
locally finite cylindric algebras. Moreover, its subcategory with
objects all largest locally finite subalgebras of full
\(\omega\)-dimensional cylindric set algebras is the category of
algebras that was used in the categorical algebraization of a
version of the institution of first-order logic without terms
presented in previous work by the author.