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My understanding is that a self-hosting compiler will take as input the language it is written in and a cross-compiler will output a language which runs on a platform which is different to what it runs on.

Can we combine these two compiler types into one?

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    $\begingroup$ A compiler, of any kind, can be written in any language. So a cross-compiler can be written in C, Java, Haskell, Prolog, etc... That includes cross-compilers which output to different platforms than the one they are written on. So yes, you can. These features are completely orthogonal. $\endgroup$ – Jake Apr 26 '16 at 15:49
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This is certainly possible, and there are two possibilities:

  1. There are two compilers for a language. For example, Haskell has GHC, and GHCJS, which is a JavaScript cross-compiler. GHC is written in Haskell, and GHCJS is written in Haskell (and normally compiled with GHC), so GHCJS is a self-hosted cross-compiler.

  2. You run your self-hosted cross-compiler in a target language. So, if you had a Foo to Python cross-compiler, and it's the only compiler for Foo, and it's written in Foo, it can be self hosting, but you would have to cross-compile it, then run the result in a Python interpreter.

Basically, be careful that you don't confuse a language with a compiler. A language can have many compilers.

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