Consider a product of projective varieties $X\times Y$ and two cohernet sheaves $\mathcal{F},\mathcal{G}$ such that
$$\mathcal{F}|\_{x\times Y}\cong\mathcal{G}|\_{x\times Y}$$
for any $x\in X$. Thanks to user 40297, we know that in general we do not have $\mathcal{F}\cong q^\*\mathcal{M}\otimes\mathcal{G}$ for some line bundle $\mathcal{M}$. Now I wonder, will we have $$q^\*\mathcal{M}\otimes\mathcal{F}\cong q^\*\mathcal{N}\otimes\mathcal{G}$$
for some locally free sheaves $\mathcal{M},\mathcal{N}$ on $X$ of the same rank?



Does $\mathcal{F}|_{x\times Y}\cong\mathcal{G}|_{x\times Y}\Rightarrow q^*\mathcal{M}\otimes\mathcal{F}\cong q^*\mathcal{N}\otimes\mathcal{G}$
No. For instance, take $Y$ to be a point, $X = \mathbb{P^1}$, $F = \mathcal{O} \oplus \mathcal{O}(1)$, and $G = \mathcal{O} \oplus \mathcal{O}$.


