Gelfand and Manin give the definition of an internal $Hom^{\bullet}(A^{\bullet}, B^{\bullet})$ complex for two cochain complexes $A^{\bullet}, B^{\bullet}$:
$$Hom^n (A^{\bullet}, B^{\bullet})=\prod_{i} Hom (A^i, B^{i+n}).$$
Then in exercise IV.2.2.b as a hint for a construction of a spectral sequence they ask to take an internal Hom complex for two filtered complexes $(A^{\bullet} \supset ... \supset F^p A^{\bullet} \supset ... )$ and$(B^{\bullet} \supset ... \supset F^q B^{\bullet} \supset ...)$, with the induced filtration.
So how do I define $$(F^p Hom^{\bullet}(A^{\bullet}, B^{\bullet}))^n \ \ ?$$
I am struggling to set up the correct indexing for this natural filtration, and usually it helps to look at the easier tensor product case first, but I don't see how translate it to our case here (because of the contravariance in the first argument).
Edit: The following filtration on the level of the double complex seems to work: $$F^p Hom (A^n, B^m) = \{ f: A^n \to B^m | f(F^k A^n) \subset F^{k+p} B^m \text{ for every} \ k \}.$$ I'll leave the question in case someone finds it useful later on.