In a small saucepan, over low to medium heat, soften blackberries and strawberries in a few tablespoons of water. Let them come to a simmer, mash with a potato masher. Push the berry juices through a fine strainer. Discard the pulp and seeds.

In a gallon pouring pitcher, add the simple syrup, lemon juice, cold water and berry juices. Add plenty of ice, stir and enjoy.