Do the Colosseum and Theatre need separate population for their benefits?

It is stated for both the Colosseum and the Theatre that they can only provide as much happiness as there is population. I believe they both provide +5 Happiness for a population of 5.

Do they stack on the same population meaning I would get 10 Happiness with a population of 5?

Solution 1:

No, the population limit applies to the total happiness from all buildings. Theater provides 4 happiness and Colosseum provides 3 happiness; if you have a size 6 city with both buildings, you would get a total of 6 happiness. You would need a size 7 city to get the full benefit from them.