What is the benefit or purpose of having different hemispheres of the brain contribute more to different cognitive processes?
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First, biological features do not always arise from some intrinsic benefit. They can also be byproducts of other adaptations, or spandrels.
That being said, one example of a possible benefit is specialization. For instance, birds will develop asymmetries in their visual system based on light inputs to their outward-facing eye (one eye faces the eggshell, the other is obscured within the egg). Distinct asymmetries in the brain allow birds to use one eye or brain region for finding and tracking prey (a very robust detail-oriented object identifier), and the other will be turned towards the sky, looking out for predators (and isn't as detail-oriented).