Total Hom complexes preserve chain homotopies

Suppose \(C_*\) and \(D_*\) are bounded-below chain complexes. Form the double complex having \(\operatorname{Hom}(C_i,D_j)\) in bidegree \((i,j)\), and write \[ \underline{\operatorname{Hom}}(C_*,D_*) \] for its total complex. The following facts are given:

1. If \(C_*\) is free, then the functor \[ D_*\longmapsto \underline{\operatorname{Hom}}(C_*,D_*) \] preserves all quasi-isomorphisms.

2. For any \(D_*\), the functor \[ C_*\longmapsto \underline{\operatorname{Hom}}(C_*,D_*) \] preserves chain homotopies.

Prove (2).

If \[ H_0(C)=\mathbb Z,\qquad H_1(C)=\mathbb Z/3\oplus\mathbb Z,\qquad H_2(C)=\mathbb Z/4,\qquad H_3(C)=\mathbb Z/2, \] and \[ H_0(D)=\mathbb Z/8,\qquad H_1(D)=\mathbb Z\oplus\mathbb Z/6,\qquad H_2(D)=\mathbb Z/2, \] with all other homology groups zero, determine the homology groups of \(\underline{\operatorname{Hom}}(C_*,D_*)\). Use the same circle of ideas used in proving the Künneth theorem.