I'm confused about Hatcher's Theorem 3.5 on page 203. The theorem states:
$H^n(X;G) \approx$ ker$d_n/$ Im$d_{n-1}$. Furthermore, the cellular cochain complex $\{H^{n}(X^n,X^{n-1};G),d_n\}$ is isomorphic to the dual of the cellular chain complex, obtained by applying Hom$(-,G)$.
I am particularly confused what it means for the cellular cochain complex to be isomorphic to the dual of the cellular chain complex.
On
Given a chain complex $(C_i(X), \partial_i)$, there are two ways to think of Cellular Co-Chain complex. The first way is by constructing the cellular chain complex $C^{Cell}_i(X) = H^i(X^{i},X^{i-1};\mathbb{Z})$ with boundary maps $D_i = j_{i-1} \circ \partial_i$, ie boundary maps induced by the provided boundary maps $\partial_i$. Now we could take take the cohomology with $G$ coefficients. So we would get $(C^{i}_{Cell}(X),d^i) = \left(\mathrm{Hom}(H_i(X^{i},X^{i-1}),G),D_i^{*} \right)$. The second way to think about it would be to directly take the cohomology of the pair $(X^{i},X^{i-1})$ in $G$ coeffients, that is defining $(C^{i}_{Cell}(X),d^i) = \left((H^{i}(X^{i},X^{i-1};G),\delta^i \circ j_i\right)$, where the coboundary maps are induced by the coboundary maps on the original chain complex
The theorem states that these two ways of defining the cellular cochain complex are isomorphic as co chain complexes. This follows from the universal coefficient theorem
Well, there is the cohomology complex with maps $$\delta:H^i(X^{i},X^{i-1}) \to H^{i+1}(X^{i+1},X^i)$$ and there is also the complex with maps $$d^*:\mathrm{Hom}(H_i(X),G) \to \mathrm{Hom}(H_{i+1}(X),G)$$ where $d^*(f)=f \circ d$.
To show an isomorphism, you want to exhibit a family of maps $$f_i:H^i(X^{i},X^{i-1}) \to \mathrm{Hom}(H_i(X),G)$$ where $f_{i+1} \circ \delta=d^* f_{i}$ (so that it commutes with the maps.)