Internal Hom of chain complexes

Definition 6.6.29 (Internal hom of chain complexes). The internal hom of chain complexes \(C_{\bullet }, D_{\bullet } \in \Ch (\Aa )\) is the chain complex \(\uHom (C, D)_{\bullet }\) defined by \[ \uHom (C, D)_n \,:=\, \prod _{p \in \Z } [C_p, D_{p+n}] \] with differential \((d\phi )_p = d \circ \phi _p - (-1)^n \phi _{p-1} \circ d\) for \(\phi = (\phi _p)_p \in \uHom (C, D)_n\).