Place the cream of coconut, coconut milk, sweetened condensed milk and evaporated milk in a blender along with the vanilla extract, ground nutmeg and ground cinnamon. Blend together for a minute until well-combined.
Add the rum and blend well. Taste the coquito and adjust with more rum as needed.
Pour the mixture carefully into a large enough container (it can be a pitcher or a glass bottle) and throw in the whole cinnamon sticks.
Cover the container and place in the fridge for at least 1 hour. Doing this infuses the coquito with the flavors from the cinnamon sticks and makes it all the more delicious!
After 1 hour you can take it out of the fridge to serve or divide into smaller bottles to gift your loved ones.
If serving, let the coquito return to room temperature first, before pouring it out and serving it either neat or over ice with a sprinkling of cinnamon on top.