  We generalise the construction of the Lie algebroid of a Lie groupoid so that
it can be carried out in any tangent category. First we reconstruct the
bijection between left invariant vector fields and source constant tangent
vectors based at an identity element for a groupoid in a category equipped with
an endofunctor that has a retraction onto the identity functor. Second we use
the full structure of a tangent category to construct the algebroid of a
groupoid. Finally we show how the classical result concerning the splitting of
the tangent bundle of a Lie group can be carried out for any pregroupoid.
