I understand the proof of Horseshoe lemma as it is presented in, e.g. Weibel's book. However both Weibel and these notes note an additional property which is at the bottom of my screenshots here:
This property is used in a crucial way in the proof that left derived functors form a homological $\delta$-functor in both said notes and Weibel's book. How does it follow from the Horseshoe Lemma?
I'm quite ashamed for asking this since the answer turnes out to be trivial, but I suppose if it's trivial, then why don't write it down.
First, in every additive category $\mathsf{A}$, every morphism between biproducts $f\colon A_1\oplus A_2 \to B_1\oplus B_2$ can be written uniquely in the following form:
$$f = \begin{bmatrix}f_{11} & f_{12} \\ f_{21} & f_{22} \end{bmatrix}$$
where $f_{ij}$ is the morphism $A_j\to B_i$ such that $$f_{ij} = \pi^B_i\circ \begin{bmatrix}f_{11} & f_{12} \\ f_{21} & f_{22} \end{bmatrix} \circ l^A_j$$ where $\pi^B_i\colon B_1\oplus B_2 \to B_i$ are product projections and $l^A_j\colon A_j\to A_1\oplus A_2$ are coproduct injections.
Moreover, we have $\pi^A_i\circ l^A_j$ equal to $1_{A_i}$ if $i = j$ and to $0$ otherwise (and similarly for $B$).
Now consider the (commutative by the Horseshoe lemma) diagram:
and set
$$d_{n + 1} = \begin{bmatrix} f_{11} & f_{12} \\ f_{21} & f_{22} \end{bmatrix}$$
We compute, by the commutativity of the diagram:
$$f_{11} = \pi'_n\circ d_{n + 1}\circ l'_{n + 1} = \pi'_n\circ l'_n\circ d'_{n + 1} = d'_{n + 1},$$ $$f_{21} = \pi''_n\circ d_{n + 1} \circ l'_{n + 1} = d''_{n + 1}\circ \pi''_{n + 1}\circ l'_{n + 1} = d''_{n + 1}\circ 0 = 0,$$ $$f_{22} = \pi''_n\circ d_{n+1}\circ l''_{n + 1} = d''_{n + 1}\circ \pi''_{n + 1}\circ l''_{n + 1} = d''_{n + 1}.$$
Then $f_{12}$ is the desired morphism $\lambda_{n + 1}$.