  Grothendieck fibrations provide a unifying algebraic framework that underlies
the treatment of various form of logics, such as first order logic, higher
order logics and dependent type theories. In the categorical approach to logic
proposed by Lawvere, which systematically uses adjoints to describe the logical
operations, equality is presented in the form of a left adjoint to reindexing
along a diagonal arrows in the base. Taking advantage of the modular
perspective provided by category theory, one can look at those Grothendieck
fibrations which sustain just the structure of equality, the so-called
elementary fibrations, aka fibrations with equality. The present paper provides
a characterisation of elementary fibrations based on particular structures in
the fibres, called transporters. The characterisation is a substantial
generalisation of the one already available for faithful fibrations. There is a
close resemblance between transporters and the structures used in the semantics
of the identity type of Martin-L\"of type theory. We close the paper by
comparing the two.
