Evaluation Map of Line Bundle from Lazarsfeld I

Question

In Chapter 1.1B, page 13, of Lazarsfeld's book Positivity in Algebraic Geometry I, he considers the following map.

Let $L$ be a line bundle on a variety $X$ and $V\subset H^0(X, L)$ be a nonzero finite dimensional subspace. Then he writes: "Evaluation of sections in $V$ gives rise to a morphism $$\text{eval}_V:V\otimes_{\mathbb{C}}\mathcal{O}_X\to L$$ of vector bundles on $X$."

My question is - what exactly is this map, and why is called the "evaluation" map? My guess is that over any open $U$, the map is just $s|_U\otimes t|_U\mapsto t|_U\cdot s|_U$ and this indeed seems to be the case in at least Cannonical evaluation map. However, what I don't see is why this is called the "evaluation" map. To conclude, my questions are as follows:

1. What is this map - is what I've written correct?
2. Why is this map called the "evaluation" map?

Thanks.

Answer 1

The map at point $x \in X$ is defined by taking $v \otimes 1$, where $v \in V$ is considered as a global section of $L$, to the value of $v$ at $x$. By this reason it is called evaluation.