  The core diagram of a double Lie algebroid consists in the core of the double
Lie algebroid, together with the two core-anchor maps to the sides of the
double Lie algebroid. If these two core anchors are surjective, then the double
Lie algebroid and its core diagram are called transitive. This paper
establishes an equivalence between transitive double Lie algebroids, and
transitive core diagrams over a fixed base manifold. In other words, it proves
that a transitive double Lie algebroid is completely determined by its core
diagram. The comma double Lie algebroid associated to a morphism of Lie
algebroids is defined. If the latter morphism is one of the core-anchors of a
transitive core diagram, then the comma double algebroid can be quotiented out
by the second core-anchor, yielding a transitive double Lie algebroid, which is
the one that is equivalent to the transitive core diagram. Brown's and
Mackenzie's equivalence of transitive core diagrams (of Lie groupoids) with
transitive double Lie groupoids is then used in order to show that a transitive
double Lie algebroid with integrable sides and core is automatically integrable
to a transitive double Lie groupoid. This research was a joint project with the
sadly deceased second author. This paper is dedicated to his memory.
