You can water lawns in the sun but it's not a good thing to do if you're looking to get and grow a healthy lawn.
If the sun is not too hot outside such as it just being in the 80 F or below range in temperature outside then it's okay to water the lawn in the sun.
However if the sun is out and the temperature outside is above 80 F or scorching hot outside then it's not good to water the lawn during the sun.
If you do water the lawn in the hot sun your lawn can get burnt and also the water will mostly evaporate from the lawn.
If you're watering your lawn in the day then the best time of day to water the lawn is early in the morning as the sun is coming up.
Or at least water the lawn before noon and before it gets too hot out.
Another good time to water the lawn is just before dark or during the night when the sun has gone down.
This will prevent the water from evaporating from the lawn and also prevent the lawn from getting scorched.
When watering grass it's best to water the grass either early in the morning or during the night or just before dark.
Especially if the weather is hot out during the day as the heat can evaporate the water you're putting on the grass.
The grass can also get burnt when watering the grass during the heat of the day so it's better to water the grass either early in the morning or late at night for the best healthy lawn.
I always run my sprinklers some early in the morning and then I water the grass again just as the nighttime is getting close.
The same is true for gardens either water the garden early in the morning or at night or before dark as the weather cools down.