nLab
Makkai duality

• M. Makkai, Stone duality for first order logic, Proceedings of the Herbrand symposium (Marseilles, 1981), 217–232, Stud. Logic Found. Math. 107, North-Holland, Amsterdam, 1982, doi; Stone duality for first-order logic, Adv. Math. 65 (1987) no. 2, 97–170, doi, MR89h:03067; Duality and definability in first order logic, Mem. Amer. Math. Soc. 105 (1993), no. 503

Some more general variants are achieved in

• Marek W. Zawadowski, Descent and duality, Annals of Pure and Applied Logic 71, n.2 (1995), 131–188
• Henrik Forssell, First-order logical duality, Ph.D. thesis, Carnegie Mellon U. 2008, pdf