  We construct a canonical correspondence from a wide class of reproducing
kernels on infinite-dimensional Hermitian vector bundles to linear connections
on these bundles. The linear connection in question is obtained through a
pull-back operation involving the tautological universal bundle and the
classifying morphism of the input kernel. The aforementioned correspondence
turns out to be a canonical functor between categories of kernels and linear
connections. A number of examples of linear connections including the ones
associated to classical kernels, homogeneous reproducing kernels and kernels
occurring in the dilation theory for completely positive maps are given,
together with their covariant derivatives.
