I am trying to understand "de Cataldo, Migliorini. The perverse filtration and the Lefschetz hyperplane theorem. *Annals of Mathematics*, 171(2010), 2089-2113." My question is about one detail in the paper. Here is my reproduction of the situation.

Let $\Sigma$ be a stratification of $\mathbb{P}^N$ adapted to a bounded complex $K$ with $\Sigma$-constructible cohomology. Let $\Sigma$ be also adapted to the embedding $Y \subseteq \mathbb{P}^N$ of an affine variety $Y$. Let $\Lambda \subset \mathbb{P}^N$ be a hyperplane, $H = \Lambda \cap Y$ and $\bar{H} = \Lambda \cap \bar{Y}$. Set $\bar{U} = \bar{Y} \setminus \bar{H}$ and $U = Y \setminus H$. Consider the cartesian diagram

$$\begin{array}[c]{ccccc} H&{\xrightarrow{i}}&Y&{\xleftarrow{j}}&U\\ \downarrow\scriptstyle{J}&&\downarrow\scriptstyle{J} && \downarrow\scriptstyle{J}\\ \bar{H}&{\xrightarrow{i}}&\bar{Y}&{\xleftarrow{j}}&\bar{U} \end{array}$$

Now the two authors stated (page 2101, following the same diagram) "By the octahedron axiom, the map $J_!j_*j^*K \to j_*J_!j^*K$ is an isomorphism if and only if the natural base change map $i^*J_*K \to J_*i^*K$ is an isomorphism."

My question is: what is exactly used about the octahedron axiom for the statement?

Thanks!