Coproducts internal to profinite spaces

We give a categorical explanation for many properties of profinite coproducts of profinite groups, which were previously proven on a case-by-case basis. All of these properties take the form "certain functors preserve profinite coproducts". We give various examples to show how our framework can be applied. We also point out connections to internal category theory and profinite Bass-Serre theory.