Let's consider a continuous map $f: X \rightarrow Y$, so that $Y= U_{1} \cup U_{2}$ -- covering by open sets and $X = f^{-1}(U_{1}) \cup f^{-1}(U_{2}) = V_{1} \cup V_{2}$. How to prove that the following diagram commutes?
$$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{llllllllllll} \ldots& \ra{\partial_{n}} & C_{n}(V_{1} \cap V_{2}) & \ra{\varphi_{n}} & C_{n}(V_{1}) \oplus C_{n}(V_{2}) & \ra{\psi_{n}} & C_{n}(X) & \ra{\partial_{n-1}} & C_{n-1}(V_{1} \cap V_{2}) & \ra{\varphi_{n-1}} & \ldots \\ & & \da{f^{n}_{12}} & & \da{f^{n}_{1} \oplus f^{n}_{2}} & & \da{ f^{n}} & & \da{f^{n}_{12}} & & \\ \ldots & \ra{\tilde{\partial_{n-1}}} & C_{n}(U_{1} \cap U_{2}) & \ra{\tilde{\varphi_{n}}} & C_{n}(U_{1}) \oplus C_{n}(U_{2}) & \ra{\tilde{\psi_{n}}} & C_{n}(Y) & \ra{\tilde{\partial_{n-1}}} & C_{n-1}(U_{1} \cap U_{2}) & \ra{\tilde{\varphi_{n-1}}} & \ldots \\ \end{array} $$
Here, $$f_{1} = f|_{V_{1}} : V_{1} \rightarrow V_{1}$$ $$f_{2} = f|_{V_{2}}: V_{2} \rightarrow V_{2}$$ $$f_{12} = f |_{V_{1} \cap V_{2}}: V_{1} \cap V_{2} \rightarrow U_{1} \cap U_{2}$$ $$\varphi: C_{n}(U_{1} \cap U_{2}) \rightarrow C_{n}(U_{1}) \oplus C_{n}(U_{2}), \varphi(x) = (x, -x)$$ $$\psi: C_{n}(U_{1}) \oplus C_{n}(U_{2}) \rightarrow C_{n}(Y), \psi(x, y) = x+ y$$
So, let's try proving that $f^{n}_{1} \oplus f^{n}_{2} \circ \varphi^{n} = \tilde{\varphi^{n}} \circ f^{n}_{12}$. So, take $x \in C_{n}(U_{1} \cap U_{2})$, then $\varphi_{n}(x) = (x, -x)$, then $f_{1}^{n} \oplus f_{2}^{n}(x, -x) = (f_{1}(x), f_{2}(-x))$, on the other hand $f_{12}(x) = f(x)$, then $\tilde{\varphi_{n}(f(x))} = (f(x), -f(x))$, so if the diagram commutes, then $f$ is odd, i.e. $f(-x)= -f(x)$. What's the problem hidden there? In general it's not required from $f$ to be an odd function.
Also, it can be shown that these long sequences are exact (it's always proven while proving that the Mayer-Vietoris sequence is exact). Does the feature of exactness will help much in this concrete case?
Any help would be much appreciated.