Yes $ R $ is a commutative ring, $ ^ cdot, Y ^ cdot $ are complexes of $ R $ modules, it is natural to define $ X ^ cdot otimes ^ cdot Y ^ cdot $ by the nth component $ coprod_ {i + j = n} X ^ i otimes_R Y ^ j $ with differential:$ d (x otimes y) = dx otimes y + (- 1) ^ { mid x mid} $ otimes $and there is a deputy $ (X ^ cdot otimes ^ cdot-, Hom ^ cdot (X ^ cdot, -) $ between DG category $ C_ {d}} (R) $ and $ C_ {d}} (R) $.

I want to find the graduated version of the tensor and hom complexes of graduated module complexes.

Yes $ A $ is a commutative graduated ring. $ A-Gr $ graduation category $ A $ modules with homomorphism homogeneous morphism $ A $-modules.Let $ M, N $ be ranked two $ A- $modules. It is also natural to define $ M otimes ^ * N = coprod _n (M ototimes ^ * N) ^ n $ with n-th component is generated by $ x otimes y $ such as $ mid x mid + mid y mid = n $.To define $ ^ * Hom (M, N) = coprod_n Hom_ {A-Gr} (M, N)[n]$.

If to consider $ A = A ^ 0 $,then $ Hom_ {A-Gr} (M, N) = prod_nHom_A (M ^ n, N ^ n) $, direct calculation impossible $ (M otimes ^ * -, ^ * Hom (M, -)) $ is an assistant couple.**So I think that in general, it is not assistant, right?**

Consider complexes of $ A $ modules $ X ^ cdot, Y ^ cdot $How to define $ X ^ cdot otimes ^ cdot Y ^ cdot $ and $ Hom ^ cdot (X ^ cdot, Y ^ cdot) $?

Similarly, define $ (X ^ cdot otimes ^ cdot Y cdot) ^ n = coprod_ {i + j = n} X ^ i otimes ^ * Y ^ j $but how to define the differential? Yes $ x in (X ^ i) _j $,or $ X ^ i $ is a noted module. There are two different ways of defining $ d (x otimes y) = dx otimes y + (- 1) ^ {i} x otimes dy $ or replaced $ (- 1) ^ i $ by $ (- 1) ^ {i + j} $Both cases can make it complex.It seems that $ (- 1) ^ {i + j} $ Maybe better. What is the standard way? that is to say what is the degree of $ x $? How to define the graduated version differential of the hom complex?

Are there good literatures on the class of graduated module complexes? With homological details.

Thank you in advance.