Let $X$ be a smooth projective surface over $\mathbb{C}$. Let $L$ be an ample line bundle on $X$. Let $F$ be a $\mu_L$ semistable rank 2 vector bundle on $X$ (semistability in the sense of Mumford-Takemoto).

We have the projections $p_i:X\times X\longrightarrow X$ for $i=1,2$. Then $L'=p_1^*L\otimes p_2^*L$ is an ample line bundle on $X\times X$.

Are (a) $p_1^*F\oplus p_2^*F$ and (b) $p_1^*F\otimes p_2^*F$ are $\mu_{L'}$-semistable on $X\times X$.

I was first wondering if the pull back $p_i^*F$ is $\mu_{L'}$- semistable. Maruyama's result holds only for finite morphism (pullback of a $\mu_L$-semistable vector bundle is $\mu_{g^*L}$-semistable if $g$ is a finite morphism). Is there some result I can use?

Thanks in advance!