It is established that if
is a Banach space, then the composition operator on
-valued Hardy spaces, weighted Bergman spaces, and on Bloch spaces is weakly compact, respectively Rosenthal (i.e., strictly cosingular) if and only if both the identity operator on
and the corresponding composition operator on the analogous scalar valued spaces have this same property. The final section of the paper contains some results for ‘general vector-valued spaces’, where the range space is defined in terms of a Banach space of analytic functions on the unit disc whose closed unit ball is compact in the compact open topology.