I'm struggling to understand part of the descripition of a connection on page 167 of Madsen, Tornehave "From Calculus to Cohomology". Immediately after defining a connection $\nabla$ on a bundle $\xi$ over $M$, they write:
Let $\tau$ be the tangent bundle of $M$. Then $\Omega^1(M) = \Omega^0(\tau^*)$, and by Theorem 16.13 we have the following rewritings of the range of $\nabla$, $$\Omega^1(M) \otimes_{\Omega^0(M)} \Omega^0(\xi) \cong \Omega^0(\operatorname{Hom}(\tau, \xi)) \cong \operatorname{Hom}_{\Omega^0(M)}(\Omega^0(\tau),\Omega^0(\xi))$$ A tangent vector field $X$ on $M$ is a section in the tangent bundle $X \in \Omega^0(\tau)$, and induces an $\Omega^0(M)$-linear map $\operatorname{Ev}_X : \Omega^1(M) \to \Omega^0(M)$, and hence an $\Omega^0(M)$-linear map $$\operatorname{Ev}_X : \Omega^1(M) \otimes_{\Omega^0(M)} \Omega^0(\xi) \to \Omega^0(\xi).$$
Here $\Omega^k(M)$ is the space of $k$-differential forms on $M$ and $\Omega^0(\xi)$ is the space of sections on $M$.
I'm having a problem with the second part (from "A tangent vector field...). Specifically, I don't understand how the vector field induces the first map. Any help would be appreciated.
$\Omega ^1(M)$ is the space of differential one form, which is the set of smooth map
$$\alpha : M \to T^*M$$
so that $\pi \circ \alpha = id_M$. For each vector fields $X$, the map is
$$\text{Ex}_X : \Omega^1(M) \to \Omega^0(M),\ \ \ \text{Ex}_X (\alpha)(x) = \alpha_x(X)$$