See Differentiable function

Between differentiable manifolds

$f: X \to Y$ between two Differentiable manifolds

$f$ when pulled back to the open sets of charts, is smooth in the usual sense of real analysis.

Functorial preperties

Smoothness is preserved under composition

the identity map is smooth

Therefore they form a Category

Manifolds and smooth maps behave nicel under products

X,Y manifolds. then $X \times Y$ has a natural manifold structure, given by the Product topology.The dimensions add.

If $f: X \to Y$, $g: X\to Z$ are smooth, the direct product $(f,g): X \to Y \times Z$ ... is smooth

If $f: W \to Y$, $g: X \to Z$ are smooth, the prdouct $f \times g: W \times X \to Y \times Z$ ... $(w,x) \mapsto (f(w),g(x))$ is smooth