The three definitions of Cech cohomology (simplical complex vs. presheaf vs. sheaf)

Question

I came across the following three definitions of Cech cohomology group of a topological space $X$:

1. [Source: Munkres, Elements of Algebraic Topology, pp. 437]. The Cech cohomology group of $X$ in dimension $k$, with coefficients in the abelian group $G$ is $$\check{H}^k(X,G) =\lim_{\substack{\longrightarrow\\\mathcal{U}\in A}} H^k(N(\mathcal{U}), G)$$ where $A$ is the directed set consisting of all open coverings of the space $X$, directed by letting $\mathcal{U} < \mathcal{V}$ if $\mathcal{V}$ is a refinement of $\mathcal{U}$, and $N(\mathcal{U})$ is an abstract simplicial complex called nerve of $\mathcal{U}$.
2. [Source: Bott and Tu, Differential Forms in Algebraic Topology, pp. 112] The Cech cohomology of $X$ with values in the presheaf $\mathscr{F}$ is $$\check{H}^k(X,\mathscr{F}) =\lim_{\substack{\longrightarrow\\\mathscr{U}\in A}} H^k(\mathcal{U}, \mathscr{F})$$ where $A$ is the directed set consisting of all open coverings of the space $X$, directed by letting $\mathscr{U} < \mathscr{V}$ if $\mathscr{V}$ is a refinement of $\mathscr{U}$.
3. [Source: Miranda, Algebraic Curves and Riemann Surfaces, pp. 296] The $k^{th}$ Cech cohomology group of a sheaf $\mathscr{F}$ on $X$, for $k\geq 0$ is $$\check{H}^k(X,\mathscr{F}) =\lim_{\substack{\longrightarrow\\\mathscr{U}\in A}} H^n(\mathcal{U}, \mathscr{F})$$ where $A$ is the directed set consisting of all open coverings of the space $X$, directed by letting $\mathscr{U} < \mathscr{V}$ if $\mathscr{V}$ is a refinement of $\mathscr{U}$.

I have seen the proof of isomorphism between Cech and de Rham cohomology using all these definitions of Cech cohomology. Depending on the definition, we have to use different kinds of tools for the proof.

How are all these definitions equivalent? Is there any reference which discusses the equivalence of these definitions? Are all these definitions special case of a general definition of Cech cohomology group?