No, he shouldn't continue lifting this way. When you're lifting something, you need to use proper body mechanics. With the way he stooped over to pick up the box, he has a high chance of injuring his back; you shouldn't bend at your waist when picking something up. Also, you should hold the object close to your body at your waist level. In this case, he should stand with his feet shoulder-width apart, he should bend at his knees instead of his waist, and he should be in a squat-like position when picking up the box.
