Slice the cabbage in half and remove the core from each section.
Using a sharp knife, cut the cabbage into very fine strips, then chop further into bite-sized pieces. Transfer to a large mixing bowl.
Shred the carrot finely, using either a box grater or a food processor. Add the shredded carrot to the bowl with the cabbage.
In a separate bowl, whisk together the mayonnaise, granulated sugar, apple cider vinegar, lemon juice, salt, and pepper until the mixture is smooth and well-emulsified.
Pour the prepared dressing over the cabbage and carrot mixture. Stir thoroughly to ensure the vegetables are evenly coated with the dressing.
Cover and refrigerate until serving. For optimal flavor and texture, allow the coleslaw to rest for several hours or overnight, mixing occasionally.