I am working in derived category $D^b(X)$ of coherent sheaves on a smooth projective varitey. Let $E,F$ be two sheaves on $X$, with $\mathrm{R}Hom(E,F)=k\oplus k[-1]$, I consider the following triangle: $$\mathrm{RHom}(E,F)\otimes E\rightarrow F\rightarrow G$$. We know that in Grothendieck group, $[F]=[G]$ since $[\mathrm{RHom}(E,F)\otimes E]=0$. Now I consider relative situtaion. Let $X\times\mathbb{P}^1\xrightarrow{\pi}\mathbb{P}^1$ and $X\times\mathbb{P}^1\xrightarrow{q} X$ to be two projection maps. Consider the relative version of the triangle above: $$\pi^*R\pi_*\mathrm{R}\mathcal{H}om(q^*E,\mathcal{F})\otimes q^*E\rightarrow\mathcal{F}\rightarrow \mathcal{G}$$ where $\mathcal{F}$ is a coherent sheaf on $X\times\mathbb{P}^1$, $i_s^*\mathcal{F}=F$ for each $s\in\mathbb{P}^1$.
I expect in $K_0(X\times\mathbb{P}^1)$, $[\pi^*R\pi_*\mathrm{R}\mathcal{H}om(q^*E,\mathcal{F})\otimes q^*E]$ is also 0. Is this true? How do I show this if it is true?
