  In this paper, we use a categorical and functorial set up to model the syntax
and inference of logics with 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,
resulting in a much clearer framework for algebraic semantics and
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.
