Whether this takes the axiom of choice depends on exactly what version you want, and on what foundation, but basically yes the general result requires the axiom of choice. So the object part of a lim functor is a bit of a mystery as you say.
The arrow part, though, is no mystery. So it is not a question of anafunctors or pseudo-functors. Once you have chosen one limit L for a diagram F and another L' for a diagram F', then each natural transformation F-->F' induces a unique cone to F' from L and thus a unique arrow from L to L' commuting with the cones. This cone commuting condition obviously gets along with composition of natural transformations so it takes composite diagram transformations to the strict composites of the induced arrows between limits.
Pseudo-functors arise in a related stituation using fibered categories, where you sort of think of cartesian arrows as pullback diagrams yet they do not give strict functors. But in general cartesian arrows are actually not defined as pullbacks and even in cases where they are defined as pullbacks the functoriality problem is not about a functorial selection of pullbacks. It is about making pasting of pullback squares into a functor. That is another problem. When you just want to look at limits of diagrams of a given shape J in a given category C, you can choose one selected representative of each in a strictly functorial way if you want to.