Definition 2.5.2.8 (Differential Graded Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be differential graded categories. A differential graded functor $F$ from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ consists of the following data:
For each object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, an object $F(X) \in \operatorname{Ob}(\operatorname{\mathcal{D}})$.
For each pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a chain map $F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )_{\ast }$.
These data are required to satisfy the following conditions:
For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the chain map
\[ F_{X,X}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\ast } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(X) )_{\ast } \]carries the identity morphism $\operatorname{id}_{X}$ to the identity morphism $\operatorname{id}_{ F(X)}$.
For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$ and pair of morphisms $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{m}$, $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{n}$, we have $F_{X,Z}( g \circ f) = F_{Y,Z}(g) \circ F_{X,Y}(f)$.
We let $\operatorname{Cat}^{\operatorname{dg}}$ denote the category whose objects are (small) differential graded categories and whose morphisms are differential graded functors.