This is a part of statement made in Cartan Eilenberg Homological Algebra Pg 44 on Satellite.
Given $0\to A^0\to A^1\to\cdots\to A^q\to 0$, one can induce connecting homomorphism by short exact sequence by applying satellite construction as satellite functor induces long exact sequence. Say $T=\{T^n\}$ is a family of additive functor with connecting homomorphism $T^n(A'')\to T^{n+1}(A')$ for any short exact sequence $0\to A'\to A\to A''\to 0$. Consider the following commutative diagram. $0\to A'\to A\to A''\to 0$
$0\to B'\to B\to B''\to 0$
$0\to C'\to C\to C''\to 0$
There will be arrows in vertical direction s.t above diagram is commutative and exact in columns.
Then there is an anticommutative diagram resulted. The morphisms $T^{n-1}(C'')\to T^n(C')\to T^{n+1}(A')$ and $T^{n-1}(C'')\to T^n(A'')\to T^{n+1}(A')$ differ by a minus sign.
$\textbf{Q:}$ Where is the origin of this anticommutative diagram? Any particular reason to expect this? I am aware this happens in homology part. Is this hinting the bicomplex for spectral sequence?