A duality generalizing Lefschetz duality of a non-compact manifold

Question

I am trying to solve out Exercise 3.3.35 of Hatcher's Book on Algebraic Topology:

I have two basic questions:

1. How to define the long exact sequence in the first row? For example, can we first find a directed system of long exact sequences, which will be the first row after the direct limit is taken?
2. After the proceeding question is solved, how to prove that the second vertical mapping is an isomorphism? I am wondering it because it is essential to the application of the five-lemma and any other duality with boundary in the book, e.g., Lefschetz duality, does not apply to it because $M$ now is non-compact.