In John Lee's Introduction to Smooth Manifolds, Proposition 18.13's proof, the author states
"By the characterization of $\partial_*$, we can let $c=\partial f$ where $f,f'$ are smooth $p$-chains in $U$ and $V$, respectively, such that $f+f'$ represents the same homology class as $e$".
I'm guessing that there may be some implicit identifications being made here but that's not exactly what Proposition 18.13 would give us. We would have $f,f'$ smooth $p$-chains such that $k_{\#}f+\ell_{\#}f'$ represents the same homology class as $e$ and then there would exist a smooth $(p-1)$-chain $c$ such that $i_{\#}c=\partial f, j_{\#}c=-\partial f'$. I don't get how now we can say $c=\partial f$.
If we are making identifications so that we see all of the chains on $M$ (not on $U,V$, or $U\cap V$) then I think I get why we can say $c=\partial f$ but then I have a follow-up question. At the end of the proof, how exactly can we write $$\int_c \omega = \int_{\partial f} \omega?$$ Is there some implicit result here that if $U\subseteq M$ is an open subset, we have a differential form $\omega$ on $U$ and a chain $c$ on $U$, then it's the same if we integrate $\omega$ over $c$ or if we view both of these in $M$ and then integrate?
Thank you!