Short question:

Let $M$ and $N$ be smooth manifold, with appropriate smooth function algebras $C^\infty(M,\mathbb{R})$ and $C^\infty(N,\mathbb{R})$.

Can we express the smooth function algebra of the cartesian product manifold in terms of $C^\infty(M,\mathbb{R})$ and $C^\infty(N,\mathbb{R})$?

I know it is neither (equivalent to) $C^\infty(M,\mathbb{R}) \oplus C^\infty(N,\mathbb{R})$ nor $C^\infty(M,\mathbb{R})\otimes_{\mathbb{R}}C^\infty(N,\mathbb{R})$.

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A more general question is,if there is a general rule to get from categorical constructions on manifolds to constructions on the appropriate smooth function algebra.

Maybe this boils don to the question whether or not the functor $C^\infty(\cdot,\mathbb{R})$ from smooth manifolds to ass. comm. unitary $\mathbb{R}$-algebras preserves (co)limits.

That's indeed the deeper question.

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P.S.: I tagged it in particular as algebraic-geometry related, do to the category theory related part...