Quick recap
Right after Mustafar, Vader is still a new apprentice fresh in the armor, and he needs a lightsaber. Sith tradition says to take a Jedi's crystal and make it bleed, so that's what he sets out to do: hunt down a Jedi who survived Order 66. Meanwhile, the Emperor puts him to work on the machinery of the new Empire and tests how far the new Sith Lord's loyalty goes. Vader lives with the pain, the rage and the memory of Padmé, which he tries to bury. The arc ends with the crystal bled red and Vader holding the lightsaber he'll carry for the rest of his life.
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