I am a beginner in sheaf cohomology (especially on Riemann surface). Before clarifying my confusion, let us start at the direct limit, it can be obtained as a quotient of a disjoint union:$$\varinjlim_{i\in I} F_i=(\coprod_{i\in I} F_i)/\sim \tag1$$
But I can't understand this algebraic result intuitively, so I decided to skip it. But this key formula seems to connect many definitions in the next chapter.
For example, we can construct Cech cohomology group by taking the direct limit process by$$H^p(X,F)=\varinjlim_{\mathcal U} H^p(\mathcal U,F) \tag2$$
where $X$ is a topological space (especially in Riemann surface) and we have a cover $(U_i)_{i\in I}=\mathcal U$.
But we can also construct it by using the quotient of a disjoint union by applying $(1)$, that is to say:$$H^p(X,F)=(\coprod_{\mathcal U}H^p(X,F))/\sim \tag3$$
Also, we can define or construct many objects by using $(1)$, for example the germ over $p$ in a manifold as follows$$C^{\infty}(p)=\varinjlim_{U}A^0(U)=(\coprod_{U\ni p} A^0(U)/\sim \tag4$$
the last limit process can be achieved because the subset $U$ can be "finer and finer" so we can define it in a preorder way.
In the end, let me generalize my confusion and my understanding:
How to understand $(1)$ or why we can define such a limiting process by using a quotient of a disjoint union? (If $(1)$ is not a definition, then how to prove it?)
Consider the construction of the germ and the Cech cohomology group, it seems that the equivalence relation $\sim$ both have the meaning like "agree on their intersection" and the direct limit process has a meaning like "we hope subset get finer and finer or the refinement can tend to be the most precise). But I can't link them.
So how to understand them more intuitively? Hope I've made my exposition clear,thanks.