  We prove that, although it is undecidable if a subgroup fixed by an
automorphism intersects nontrivially an arbitrary subgroup of $F_n\times F_m$,
there is an algorithm that, taking as input a monomorphism and an endomorphism
of $F_n\times F_m$, decides whether their fixed subgroups intersect
nontrivially. The general case of this problem, where two arbitrary
endomorphisms are given as input remains unknown. We show that, when two
endomorphisms of a certain type are given as input, this problem is equivalent
to the Post Correspondence Problem for free groups.
