  Adopting the pullback approach to global Finsler geometry, the aim of the
present paper is to provide new intrinsic (coordinate-free) proofs of intrinsic
versions of the existence and uniqueness theorems for the Cartan and Berwald
connections on a Finsler manifold. To accomplish this, the notions of semispray
and nonlinear connection associated with a given regular connection, in the
pullback bundle, is introduced and investigated. Moreover, it is shown that for
the Cartan and Berwald connections, the associated semispray coincides with the
canonical spray and the associated nonlinear connection coincides with the
Barthel connection. An explicit intrinsic expression relating both connections
is deduced.
