# Hatcher pg 200:Clarification on duality of connecting homomorphisms

I've recently started learning algebraic topology using Hatcher's book and had a little problem with one of the proofs. In the section on relative cohomology(pg 200), Hatcher discusses a duality relationship between the connecting homomorphisms in the long exact sequence of cohomology and homology groups. The sequence is

$$\cdots \rightarrow H^{n}(X,A:G) \xrightarrow{j^{*}} H^{n}(X;G) \xrightarrow{i^{*}} H^{n}(A;G) \xrightarrow{\delta} H^{n+1}(X,A;G) \rightarrow \cdots$$ where all the maps are the obvious ones, $$\delta$$ is the connecting homomorphism.

Given the group $$H^{n}(C;G)$$, there is the natural map $$h:H^{n}(C;G) \to Hom(H_{n}(C),G)$$. Looking at the long exact sequence for relative homology where $$A \subset X$$, we get the connecting homomorphism $$\partial:H_{n+1}(X,A) \to H_{n}(A)$$. He shows that the connecting maps are dual in the sense that the following diagram commutes. Diagram

Essentially, we have to show that $$h\delta=\partial^{*}h$$. For some $$\alpha \in H^{n}(A;G)$$ represented by a cocycle $$\psi \in C^{n}(A;G)$$, $$\psi$$ can be extended to $$\bar{\psi}$$ by assigning the value 0 to singular simplices not contained in $$A$$, then composing with $$\partial:C_{n+1}(X) \to C_{n}(X)$$ gives the cochain $$\bar{\psi}\partial \in C^{n+1}(X;G)$$, now Hatcher says that this actually lies in $$C^{n+1}(X,A;G)$$. I do not understand why this is true. I feel like I'm missing something quite simple. Any help is appreciated.

The cochain $$\psi\in C^n(A)$$ is a cocycle, so it vanishes on boumdaries of chains from A. That's why when you compose it with the boundary map, it becomes a relative cocycle (= one which vanishes on boundaries from $$A$$).