This is a question about finding references and hopefully a larger context for a lemma in homological algebra I proved recently. The motivation is to understand properties of characteristic classes of $T_f$, the mapping torus of a diffeomorphism $f$ of a closed manifold, by applying the lemma to Mayer-Vietoris and a change-of-coefficients sequence for the cohomology of $T_f$.

Let $C_{ij}, 1 \leq i,j \leq 3$ be cochain complexes, and $$ \begin{matrix} & & 0 & & 0 & & 0 & & \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 & \to & C_{11} & \stackrel{g}\to & C_{21} & \stackrel{h}\to & C_{31} & \to & 0 \\ & & {\scriptstyle u}\downarrow\ & & {\scriptstyle u}\downarrow\ & & {\scriptstyle u}\downarrow\ & & \\ 0 & \to & C_{12} & \stackrel{g}\to & C_{22} & \stackrel{h}\to & C_{32} & \to & 0 \\ & & {\scriptstyle v}\downarrow\ & & {\scriptstyle v}\downarrow\ & & {\scriptstyle v}\downarrow\ & & \\ 0 & \to & C_{13} & \stackrel{g}\to & C_{23} & \stackrel{h}\to & C_{33} & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ & & 0 & & 0 & & 0 & & \end{matrix}$$

a commuting diagram where the rows and columns are short exact sequences. Let $\delta_H : H^k(C_{3j}) \to H^{k+1}(C_{1j})$ and $\delta_V : H^k(C_{i3}) \to H^{k+1}(C_{i1})$ denote the boundary homomorphisms in the associated long exact sequences. The long exact sequences can be arranged into a commuting grid

$$ \begin{matrix} H^{k-2}(C_{33}) & \stackrel{\delta_H}\to & H^{k-1}(C_{13}) & \stackrel{g}\to & H^{k-1}(C_{23}) & \stackrel{h}\to & H^{k-1}(C_{33}) & \stackrel{\delta_H}\to & H^k(C_{13}) \\ {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ \\ H^{k-1}(C_{31}) & \stackrel{\delta_H}\to & H^k(C_{11}) & \stackrel{g}\to & H^k(C_{21}) & \stackrel{h}\to & H^k(C_{31}) & \stackrel{\delta_H}\to & H^{k+1}(C_{11}) \\ {\scriptstyle u}\downarrow\ & & {\scriptstyle u}\downarrow\ & & {\scriptstyle u}\downarrow\ & & {\scriptstyle u}\downarrow\ & & {\scriptstyle u}\downarrow\ \\ H^{k-1}(C_{32}) & \stackrel{\delta_H}\to & H^k(C_{12}) & \stackrel{g}\to & H^k(C_{22}) & \stackrel{h}\to & H^k(C_{32}) & \stackrel{\delta_H}\to & H^{k+1}(C_{12})\\ {\scriptstyle v}\downarrow\ & & {\scriptstyle v}\downarrow\ & & {\scriptstyle v}\downarrow\ & & {\scriptstyle v}\downarrow\ & & {\scriptstyle v}\downarrow\ \\ H^{k-1}(C_{33}) & \stackrel{\delta_H}\to & H^k(C_{13}) & \stackrel{g}\to & H^k(C_{23}) & \stackrel{h}\to & H^k(C_{33}) & \stackrel{\delta_H}\to & H^{k+1}(C_{13}) \\ {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ & & {\scriptstyle \delta_V}\downarrow\ \ \\ H^k(C_{31}) & \stackrel{\delta_H}\to & H^{k+1}(C_{11}) & \stackrel{g}\to & H^{k+1}(C_{21}) & \stackrel{h}\to & H^{k+1}(C_{31}) & \stackrel{\delta_H}\to & H^{k+2}(C_{11}) \\ \end{matrix}$$

The grid is symmetric under translation by 3 steps up and 3 to the right.

**Lemma.** If $[\alpha] \in H^k(C_{12})$ and $[\beta] \in H^k(C_{21})$ are classes such that $g[\alpha] = u[\beta] \in H^k(C_{22})$ then there is some $[\gamma] \in H^{k-1}(C_{33})$ such that both $\delta_H[\gamma] = v[\alpha] \in H^k(C_{13})$ and $\delta_V[\gamma] = -h[\beta] \in H^k(C_{31})$.

*Proof.* Take $\chi \in C^{k-1}_{22}$ such that $d\chi = g\alpha - u\beta$. By the definition of the boundary homomorphisms, $d(v\chi) = g(v\alpha)$ implies that $\delta_H([h(v\chi)]) = [v\alpha]$, and $d(h\chi) = -u(h\beta)$ implies that $\delta_V([v(h\chi)]) = -[h\beta]$. Hence we can set $\gamma = vh\chi$.

Does this lemma look familiar? Do you know some place where it's written down?

Edit: Corrected subscripts in statement of lemma.

Update: Thanks for the alternative proofs. However, what I'm after is rather a bibliography reference that I can cite when writing up my application, just to emphasise that it is an instance of something that someone somewhere has already considered (as I imagine it is).