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In bowl of blender, add evaporated milk, cream of coconut, coconut milk, sweetened condensed milk, rum (if using), vanilla extract and ground cinnamon. Blend on high until mixture is well combined, 1-2 minutes.
Pour coconut mixture into glass bottles; cover. Transfer to refrigerator. Chill until cold.
To serve, stir or shake bottle well to combine. Pour coquito into small serving glasses. Garnish with ground cinnamon and cinnamon sticks, if desired.