Remark 2.5.4.4. Let $\operatorname{\mathcal{C}}$ be a differential graded category containing objects $X$, $Y$, and $Z$, and suppose we are given $0$-cycles $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{0}$, $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{0}$, and $h \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Z)_{0}$. Then Example 2.5.3.4 supplies an equivalence between the following data:
The datum of a homotopy from $g \circ f$ to $h$, in the sense of Definition 2.5.4.1.
The datum of a $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{\mathcal{C}})$ witnessing $h$ as a composition of $f$ and $g$, in the sense of Definition 1.4.4.1.
In particular, $h$ is homotopic to the composition $g \circ f$ (in the differential graded category $\operatorname{\mathcal{C}}$) if and only if it is a composition of $g$ and $f$ (in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$).