There is a large amount of literature on the topic of covarieties,
coequations and coequational specifications, dating back to the early
seventies. Nevertheless, coequations have not (yet) emerged as an everyday
practical specification formalism for computer scientists. In this review
paper, we argue that this is partly due to the multitude of syntaxes for
writing down coequations, which seems to have led to some confusion about what
coequations are and what they are for. By surveying the literature, we identify
four types of syntaxes: coequations-as-corelations, coequations-as-predicates,
coequations-as-equations, and coequations-as-modal-formulas. We present each of
these in a tutorial fashion, relate them to each other, and discuss their
respective uses
