= Graded Hom complex of chain complexes
{title2=$M_j(C,C')=\prod_p\operatorname{Hom}_R(C_p,C'_{p+j})$}
= Internal Hom complex
{synonym}
A graded <Hom functor> construction consists of maps shifting degree by $j$. One homological convention is $d_M f=f d_C+(-1)^{j-1}d_{C'}f$. Its degree-zero cycles are <chain maps>, and its degree-zero boundaries are null-homotopic <chain maps>, so its zeroth <homology> is the module of <chain homotopy> classes. This convention differs by degree-dependent signs from the alternative convention $d'f-(-1)^jfd$.
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