I've come across a definition for smooth maps, I'm not familiar with and I'm wondering if it's equivalent to what I'm aware of.

From Lee: a map $F : M \to N$ between manifolds is smooth if for every $p \in M$, there exist charts $(\varphi, U)$ containing $p$ and $(\psi, V)$ containing $F(p)$ such that $F(U) \subseteq V$ and the composite map $\psi \circ F \circ \varphi^{-1} : \varphi(U) \to \psi(V)$ is smooth (in the standard calculus sense).

A morphism of functionally structured spaces $(X, \mathcal F_X) \to (Y, \mathcal F_Y)$ is a map $\varphi : X \to Y$ such that composition $f \mapsto f \circ \varphi$ carries $\mathcal F_Y(U)$ into $\mathcal F_X(\varphi^{-1}(U))$.

Nevermind, I suspect these two structures are equivalent. It's just that this new definition is slightly foreign to me.

(I found the example in a book I was browsing through: it was left as an exercise to show that both are equivalent :P).

@RobertCardona It's weird to motivate smooth manifolds using sheaves. It's usually the other way around.

@BalarkaSen, that's exactly what I was trying to do. The above was Mumford's motivation for the structure sheaf. The problem is that I had never seen it this way when I studied differential geometry, so I was trying to connect the concept with what I already know, in particular, smooth maps between manifolds in terms of charts.