I am trying to get a better understanding of what Cech cohomology actually is. In particular, I want to see why it computes derived functor cohomology. I am familiar with the orthodox proof of this fact via showing that the derived functor cohomology agrees with the Cech cohomology in degree $0$, and then showing that the Cech cohomology is effaceable. But I feel that abstract proofs like that don't really show me what is going on.
So I tried to find a more concrete proof and I've been running into problems. I wanted to do this with an explicit description of the injective hull of a quasi-coherent sheaf. Hartshorne's Residues and Duality gives such a description:
For each $p \in X$, let $J(p)$ denote the injective hull of the residue field $\kappa(p)$ as an $\mathcal{O}_{X, p}$-module. Then let $\mathcal{J}_{p}$ denote the skyscraper sheaf at $p$ which associates the module $J(p)$ at the point $p$. Then $\mathcal{F}$ has an injective hull of the form, $$ \mathcal{F} \hookrightarrow \bigoplus_{p \in X} \mathcal{J}_{p} $$ where each $p$ may be repeated.
Let $\mathfrak{U} = \{ U_{\alpha} \}_{\alpha \in I}$ be an affine cover of $X$. I want to construct a morphism in degree $0$ from the Cech complex to the global sections of the injective resolution of $\mathcal{F}$, $$ \check{C}^{0}(\mathfrak{U}, \mathcal{F}) \longrightarrow \Gamma \left(X , \bigoplus_{p \in X} \mathcal{J}_{p} \right). $$ By quasi-compactness, this is equivalent to finding a morphism, $$ \bigoplus_{\alpha} \Gamma(U_{\alpha}, \mathcal{F}) \longrightarrow \bigoplus_{p \in X} \Gamma \left(X , \mathcal{J}_{p} \right). $$ Can anyone point me in the right direction? Or perhaps if this is completely the wrong way to look at things, give me some idea on how to see explicitly how the Cech cohomology maps to the derived functor cohomology on the level of cocycles and coboundaries?