In differential geometry a tangent vector at $p$ on smooth manifold $M$, is a $\Bbb{R}-$linear map $$v:C^\infty(M) \to \Bbb{R}$$
which satisfies the product rule $$v(fg) = v(f)g(p) + f(p)v(g)$$
If we consider the $C^\infty_M$ as a structure sheaf, we have the second definition a tangent vector $v$ is a $\Bbb{R}-$linear map over the stalk $$(C^\infty_M)_p\to \Bbb{R}$$ which satisfies the product rule $v(f_pg_p) = v(f_p)g_p(p) + f_p(p)v(g_p)$
I want to show these two definitions are one to one correspondence.
First given a $v$ in the first definition I want to find some morphism $$\tilde{v}:C^\infty_M\to \underline{\Bbb{R}}$$
on set of filtrant open subset around $p$, then it will naturally induce a map on the stalk. to do so , using the basic fact in differential geometry that $di_p : T_pU \to T_p M$ is isomorphism, therefore given $v$ in the first definition ,it will naturally induce a $v_{_{U}}:C^\infty(U) \to \Bbb{R}$ which satifies the product rule, therefore we define the morphism $\tilde{v}(U) = v_{_U}$, since taking stalk is functorial it will gives a map on the stalk as in the definition 2.
Conversely given a map $v'$ in the definition 2, we want to find some $v$ in the definition one, that's easy just first take $\varphi_p:C_M^\infty(M) \to C^\infty_{M,p} $ then $v := v' \circ \varphi_p$ gives the tangent vector in the first definition