Wash, remove stems, quarter and core the apples. Cook slowly with the cider and vinegar until soft.
Press the cooked fruit through a food mill or sieve. Add both sugars and spices; cook gently, stirring frequently, until thick enough to round up on a spoon.
Fill hot sterile half-pint or pint jars, leaving 1/4-inch headspace. Remove bubbles, adjust headspace, wipe rims, and apply two-piece lids.
Process half-pints or pints in a boiling-water canner for 5 minutes at 0–1,000 ft, 10 minutes at 1,001–6,000 ft, or 15 minutes above 6,000 ft.