Hi, I think it would be a good idea to activate an option in the Search Settings, so that the drop-down menu when searching shows the full path of the category, instead of just a match and its parent category. Let me explain:
If we search in the search bar for "sneakers" and it exists in two different main categories, there is no way to know which one it belongs to. For example:
Clothes and footwear > Footwear > sneakers
Babies and Children > Clothes and footwear > Footwear > sneakers

@Alej
It make sense just in case you have complex category structure and lowest levels used to repeat.
In that case best is to use hierarchy/toTree functions, but take care of performance to not overkill that.

Could you modify the code to display the icon for the corresponding category, instead of the default icon, so that the user could determine which parent category it belongs to?

As there is option to upload icon for every category and it takes few minutes to upload same icon, I do not see reason to sped time on such feature.

Hi, I'm referring to the icons that appear in the categories of the search results. Those icons should use the corresponding category icon if the icon was uploaded.

it's just general category icon.