In this paper, we use a categorical and functorial set up to model the syntax and inference of logics of algebraic signature, extending previous works on algebraisation of logics. The main feature of this work is that structurality, or invariance under substitution of variables, are modelled by functoriality in this paper, resulting in a much clearer framework for algebraisation. It also provides a very nice conceptual understanding of various existing results already established in the literatures, and derives several new results as well.
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