Why is this definition of differential for manifolds correct?

Question

The book we are using in class is Frank Warner Foundation of Differential Manifold and Lie Group. Let $M,N$ be two smooth $d$-dim manifold, the differential of a $C^\infty$ function $\phi:M\rightarrow N$ is defined by $$d\phi: M_m \rightarrow N_{\phi(m)}$$ For $v\in M_n$, and $g:N \rightarrow \mathbb{R}$ a smooth function, they define $$d\phi(v)(g) = v(g\circ \phi).$$ I fail to see why $d\phi(v)\in N_{\phi(m)}$. Since for example $v(g\circ \phi)$ would be a function from M to $\mathbb{R}$ instead of from N to $\mathbb{R}$, which would mean $d\phi(v)\in M_{m}$.$$$$Any help on understanding why $$d\phi(v)\in N_{\phi(m)}$$

Answer 1

For $v\in M_m$, $g\in C^{\infty}(N)$ and $\phi:M\to N$ you have that $v(g\circ\phi)$ is already just a number in $\mathbb{R}$, so if you define $[d\phi(v)](g)=v(g\circ\phi)$, you can see $d\phi(v)$ is a map $C^{\infty}(N)\to\mathbb{R}$.