Remark 2.5.4.8. Let $\operatorname{\mathcal{C}}$ be a differential graded category, with underlying category $\operatorname{\mathcal{C}}^{\circ }$ (Construction 2.5.2.4) and homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (Construction 2.5.4.6). There is an evident functor $\operatorname{\mathcal{C}}^{\circ } \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ which is the identity on objects, given on morphisms by the construction
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\circ }}(X,Y) = \mathrm{Z}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) \twoheadrightarrow \mathrm{H}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) = \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \quad \quad f \mapsto [f]. \]