Two semigroups are called strongly Morita equivalent if they are contained in a Morita context with unitary bi-acts and surjective
mappings. We consider the notion of context equivalence which is obtained from the notion of strong Morita equivalence by
dropping the requirement of unitariness. We show that context equivalence is an equivalence relation on the class of factorisable
semigroups and describe factorisable semigroups that are context equivalent to monoids or groups, and semigroups with weak
local units that are context equivalent to inverse semigroups, orthodox semigroups or semilattices.