In this question, $k$ is an algebraically closed field. Italic words are definitions that come from Hartshorne's Algebraic geometry section I.3 and II.1 (If you need them, leave a comment).
Part of the Exercise II.1.21(d)
Let $X=\mathbb{P}^1$ and let $\mathcal{O}$ be the sheaf of regular functions. Let $\mathcal{K}$ be the constant sheaf on $X$ associated to the function field $K$ of $X$. Show that the quotient sheaf $\mathcal{K}/\mathcal{O}$ is isomorphic to the direct sum of sheaves $\sum_{p\in X}i_P(K/\mathcal{O}_P)$ ($\mathcal{O}_P$ is the local ring of $P$), where $i_P(K/\mathcal{O}_P)$ denotes the skyscraper sheaf given by $K/\mathcal{O}_P$ at the point $P$.
Some settings
Since $X$ is irreducible, it is connected. Thus $\mathcal{K}(U)\cong K$. So I will identify $\mathcal{K}(U)$ with $K$. I will write $\langle U,f\rangle\in K$ for an element of $\mathcal{K}(U)=K$ (meaning $f:U\rightarrow k$ an regular function).
There is a natural injection $\mathcal{O}\rightarrow\mathcal{K}$ given by $\mathcal{O}(U)\ni(f:U\rightarrow k)\mapsto \langle U,f\rangle\in\mathcal{K}(U)$ so $\mathcal{K}/\mathcal{O}$ is defined well.
My problem
My strategy is finding a morphism of sheaves $\phi:\mathcal{K}\rightarrow \sum_{P\in X}i_P(K/\mathcal{O}_P)$ and use $\mathcal{K}/\ker\phi\cong\operatorname{im}\phi$. My problem occurs in defining $\phi(U):\mathcal{K}(U)\rightarrow \sum_{P\in X}i_P(K/\mathcal{O}_P)(U)=\sum_{P\in U} K/\mathcal{O}_P$. I think this map is $\langle V,f\rangle\mapsto \sum_{P\in U} \langle V,f\rangle+\mathcal{O}_P$. However, in my definition, $\langle V,f\rangle+\mathcal{O}_P=0$ for all but finitely many $P\in U$ because $\sum_{p\in X}i_P(K/\mathcal{O}_P)$ is the direct sum of sheaves (Equivalently $\langle V,f\rangle\in \mathcal{O}_P$ for all but finitely many $P\in U$.).
I wonder if there is any other way to define $\phi$ or $\sum_{P\in U} \langle V,f\rangle+\mathcal{O}_P$ is a sum of finite terms.