Algebraic characterization of first order "operations" using limit ultrapowers

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In his review http://projecteuclid.org/download/pdf_1/euclid.bams/1183537899 of the Chang and Keisler's classic book "Model Theory," M. Makkai writes: "… let us note that it is possible to formulate the results in 6.4 in a way (that the reviewer learned from A. Joyal) that brings them neatly in line with the idea of algebraic characterization of first order "operations"". My question is what could such a possible a way to reformulate the results of 6.4 be?