On p. 127 of Kashiwara-Schapira's paper "Deformation Quantization Modules", there is the following situation: $X$ is a smooth complex (quasi?)projective variety and $\delta\colon X\to X\times X$ is its diagonal embedding. The goal is to prove the Hochschild-Kostant-Rosenberg theorem in this context, but I'm having trouble understanding what's going on at one particular step.

So, Kashiwara-Schapira define a complex of $\mathcal O_{X\times X}$-modules by
$$
P_k = \delta_*\Omega_X^k\oplus\delta_*\Omega_X^{k+1}
$$
for $k\ge 0$ and $P_k=0$ for $k<0$ and with differentials being given by a composition of a projection and inclusion of factors in a direct sum. They assert that this is an exact complex away from the $0$-th term and so the canonical map $P_k\to\delta_*\mathcal O_X$ into the complex lying in degree $0$ is a quasi-isomorphism. So far so good. But then they claim that by applying functors $\delta^*$ and $H^0,$ one gets
$$
\delta^*\delta_*\mathcal O_X \to H^0(\delta^*)(P_\bullet) \simeq \bigoplus\Omega_X^i[i].
$$
**This is the point where I get lost.**
In paritcular, I don't understand where this last isomorphism comes from. I mean, if I look at
$$
\delta^*P_k=\delta^*(\delta_*\Omega_X^k\oplus\delta_*\Omega_X^{k+1}),
$$
I see no immediate way to relate this with $\Omega_X^k.$ I would maybe sooner expect to see $\delta^*\delta_*\Omega_X^k$ in its place? But really I'm just confused.

Seeing how it isn't even remarked upon in the paper, I'm probably missing a very simple point. If somebody could spell it out for me, that would be great.