A limit for each diagram need not provide a chosen limit functor without a simultaneous choice of representatives

Existence of a limit for each diagram does not itself specify one limiting cone for every diagram. The functors in Chosen limits and colimits of a fixed small shape assemble into limit and colimit functors require simultaneous choices of such cones, an application of an appropriate Axiom of Choice (The Axiom of Choice) when the diagrams and available cones form sets.

Once choices are made, no further arbitrary choices define the action on natural transformations: the universal property forces it. Different systems of chosen limits give naturally isomorphic limit functors because any two limits have unique compatible isomorphisms (Any two limits, or any two colimits, of one diagram are uniquely isomorphic compatibly with their structure maps). For a proper class of diagrams, the needed selection is a separate global or universe-level convention, not a consequence of the set-sized Axiom of Choice.