How do continuous maps between spaces induce homomorphisms between Cech cohomologies? Let $X$ and $Y$ be topological spaces and $f:X\rightarrow Y$ a continuous map. For a given open cover $\mathcal{V}$ of $Y$ and an abelian group $G$ I know how $f$ induces a group homomorphism from $H^{n}(N_{\mathcal{V}}:G)$ to $H^{n}(N_{f^{-1}(\mathcal{V})}:G)$ where $N_{\mathcal{V}}$ is the nerve of the cover $\mathcal{V}$, but how does $f$ induce a homomorphism at the level of the Cech cohomology groups $\breve{H}^{n}(Y:G)$ and $\breve{H}^{n}(X,G)$?
I've looked through a few books, but haven't been able to find what I'm looking for. Typically I only see discussions using "good covers" on smooth manifolds.
 A: Let us suppress the coefficient group $G$. For each space $X$ we obtain  the Cech system $\mathcal{C}(X)$. This is an inverse system in the homotopy category $\mathbf{HTop}$ indexed by the set $\mathfrak{U}(X)$ of open covers $\mathcal{U}$ of $X$ which is partially ordered by $\mathcal{U} \le \mathcal{V}$ if $\mathcal{V}$ refines $\mathcal{U}$. The term $X_\mathcal{U}$ is the geometric realization of the nerve $N(\mathcal{U})$ of $\mathcal{U}$, and the bonding map $p_\mathcal{U}^\mathcal{V} : X_\mathcal{V} \to X_\mathcal{U}$ is  the (uniquely determined) homotopy class of any "refinement map" $r_\mathcal{U}^\mathcal{V} : N(\mathcal{V}) \to N(\mathcal{U})$ which chooses for $V \in \mathcal{V}$ a member $r_\mathcal{U}^\mathcal{V}(V) \in \mathcal{U}$ such that  $V \subset r_\mathcal{U}^\mathcal{V}(V)$. Usually one restricts to normal covers (these have a subordinated partition of unity), but this only a technical point.
Application of $H^n$ produces a direct system $H^n(\mathcal{C}(X))$ of abelian groups whose direct limit is defined to be the $n$-th Cech cohomology group of $X$.
Each map $f : X \to Y$ induces a map of index sets $f^\# : \mathfrak{U}(Y) \to \mathfrak{U}(X)$ and a map of inverse systems $\underline{f} : \mathcal{C}(X) \to \mathcal{C}(Y)$. The latter induces $H^n(\underline{f}) : H^n(\mathcal{C}(Y)) \to H^n(\mathcal{C}(X))$. The direct limit functor gives you an induced map on the Cech cohomology groups.
A good reference is the classic
Eilenberg, Samuel, and Norman Steenrod. Foundations of algebraic topology. Princeton University Press, 2015.
