Again I'm stuck on a problem in Hartshorne! In this problem we are supposed to show that if we take the limit of all coverings in on a topological space then the Cech cohomology agrees with derived functor cohomology in degree 1. I'm stuck on part (c) but maybe my error lies in something I did in parts (a) or (b) so let me explain what I've done so far (sorry for the wall of text, I have highlighted my questions):
In part (a) we define a refinement of coverings $\mathfrak{U}=(U_i)_{i\in I}$, $\mathfrak{V}=(V_j)_{j\in J}$ to be a map $\lambda:J\to I$ of the indexing sets such that $V_j\subset U_{\lambda(j)}$. For any refinement we get a map $\breve{H}^i(\mathfrak{U},\mathcal{F})\to \breve{H}^i(\mathfrak{V},\mathcal{F})$. Indeed we have a chain map $\varphi:\breve{C}^\bullet(\mathfrak{U},\mathcal{F})\to \breve{C}^\bullet(\mathfrak{V},\mathcal{F})$ which is defined in degree $d$ by
$$\varphi(\eta)_{j_0...j_d}=\eta_{\lambda(j_0)...\lambda(j_d)}|_{V_{j_0...j_d}}.$$
In fact this map comes from a chain map of complexes of sheaves $\breve{\mathscr{C}}^\bullet(\mathfrak{U},\mathcal{F})\to \breve{\mathscr{C}}^\bullet(\mathfrak{V},\mathcal{F})$ where the degree $d$ map is defined by using the formula above on any open subset.
Then Hartshorne claims that the coverings of $X$ form a partially ordered set which I don't agree with as we can have non-identity refinements of a covering. However, the coverings of $X$ do form a small category (if we don't allow any indexing set for the coverings) and we can take the colimit of the groups $\breve{H}^i(\mathfrak{U},\mathcal{F})$
In part (b) we have to show that the natural maps $\breve{H}^i(\mathfrak{U},\mathcal{F})\to H^i(X,\mathcal{F})$ are compatible with refinements. To see this, let $\mathcal{I}^\bullet$ be an injective resolution of $\mathcal{F}$. We note that the two chain maps $\breve{\mathscr{C}}^\bullet(\mathfrak{U},\mathcal{F})\to \mathcal{I}^\bullet$ we get from the following diagram $$\breve{\mathscr{C}}^\bullet(\mathfrak{U},\mathcal{F})\to \mathcal{I}^\bullet$$ $$\downarrow{}{}\quad\quad\quad\downarrow=$$ $$\breve{\mathscr{C}}^\bullet(\mathfrak{V},\mathcal{F})\to \mathcal{I}^\bullet$$ are chain homotopic and therefore induce the same map after applying global sections and then taking homology.
Now it's part (c) that I'm stuck on. We have to show that the natural map $\lim_{\mathfrak{U}}\breve{H}^1(\mathfrak{U},\mathcal{F})\to H^1(X,\mathcal{F})$ is an isomorphism. Following the hint we embed $\mathcal{F}$ into a flasque sheaf $\mathcal{G}$ and consider the short exact sequence $$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad0\to \mathcal{F}\to \mathcal{G}\to \mathcal{R}\to 0.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(*)$$ We construct a short exact sequence of complexes of groups $$\quad\quad\quad\quad\quad\quad\quad\quad0\to \breve{C}^\bullet(\mathfrak{U},\mathcal{F})\to\breve{C}^\bullet(\mathfrak{U},\mathcal{G})\to D^\bullet(\mathfrak{U})\to 0\quad\quad\quad\quad\quad\quad\quad\quad\quad(**)$$ simply by taking $D^i(\mathfrak{U})=\breve{C}^i(\mathfrak{U},\mathcal{F})/\breve{C}^i(\mathfrak{U},\mathcal{G})$. We note that the groups $D^i$ are global sections of the presheaf $\mathcal{R'}:U\mapsto \breve{\mathscr{C}}^i(\mathfrak{U},\mathcal{F})(U)/\breve{\mathscr{C}}^i(\mathfrak{U},\mathcal{G})(U)$ and note that $\mathcal{R}$ is the sheaf associated to $\mathcal{R}'$. The canonical map $\mathcal{R}'\to \mathcal{R}$ induces a chain map $D^\bullet(\mathfrak{U})\to \breve{C}^\bullet(\mathfrak{U},\mathcal{R})$ and it induces a map in cohomology $\alpha:H^0(D^\bullet(\mathfrak{U}))\to \breve{H}^0(\mathfrak{U},\mathcal{R})=\Gamma(X,\mathcal{R})$. From the sequences $(*)$ and $(**)$ we get two long exact sequences of cohomology and the maps $\alpha$ fit into connect the two: $$0\to \Gamma(X,\mathcal{F})\to\Gamma(X,\mathcal{G})\to H^0(D^\bullet(\mathfrak{U}))\to \breve{H}^1(\mathfrak{U},\mathcal{F})\to 0$$ $$\quad\downarrow\quad\quad=\downarrow\quad\quad\quad=\downarrow\quad\quad\quad\quad\alpha\downarrow\quad\quad\quad\quad\quad\quad\quad\quad\quad$$ $$0\to\Gamma(X,\mathcal{F})\longrightarrow\Gamma(X,\mathcal{G})\longrightarrow \Gamma(X,\mathcal{R})\longrightarrow H^1(\mathcal{F})\to 0\quad.$$ From here I wanted to show that $\alpha$ is an isomorphism and that the top row remains exact in the limit which would solve the problem by the five lemma. I was able to show that $\alpha$ is surjective in the colimit but nothing else. The issue I feel like is that we are not taking colimit over a directed set so many of the good properties of direct limits that I would like to use (exactness among other things) might not hold. Is there a way to think of coverings and refinements as a poset as Hartshorne claims it is or is there some other way to solve the problem?