Consider a bundle $E \rightarrow B$ in some category. The morphisms of it to some other bundle $E' \rightarrow B'$ is simply two morphism $E \rightarrow E'$ and $B \rightarrow B'$ which make the obvious square commute.
Suppose that fibres exist, that is there is a terminal object $*$, and for any point $x:* \rightarrow B$ there is a pullback $E_x$ of $E$ to the point.
It turns out that bundle morphisms must be fibre preserving.
Now given just a morphism $E \rightarrow E'$ can we always complete it to a bundle morphism?
As far as I can see - if it isn't fibre-preserving it isn't possible. And if it is, then there is only one $B \rightarrow B'$ that make the square commute (if it exists) - is this right? I don't see anything wrong with the first statement; the second statement seems right in $Set$ - there is always a unique completion found by projecting down the fibres, but in $Top$ the unique set map may not lift to a continuous map.
(i) Can anything be said if we don't have fibres?
(ii) Can anything more be said if not only do we have fibres, but also the bundles are locally trivial, but not neccessarily with a standard fibre?