I am a bit confused about how synthetic differential geometry deals with certain compositions of smooth maps.
We have a square root function $f(x)=\sqrt{x}\colon \mathbb{R}^+\to \mathbb{R}^+$ and we have the squaring function $g(x)=x^2\colon \mathbb{R}\to \mathbb{R}^+$ as smooth functions. But the composition $f(g(x))=|x|$ is not smooth so it shouldn't exist in a setting of smooth worlds.
I think the issue is with the formation of $\mathbb{R}^+$.
I guess my main question is if there is this composition $f\circ g$ and if not, exactly why?