Remark 2.5.2.3. Let $\operatorname{\mathcal{C}}$ be a differential graded category containing a pair of morphisms $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{m}$ and $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{n}$. It follows from the Leibniz rule
\[ \partial (g \circ f) = (\partial g) \circ f + (-1)^{n} g \circ (\partial f) \]
that if $f$ and $g$ are cycles (that is, if they satisfy $\partial f = 0$ and $\partial g =0$), then $g \circ f$ is also a cycle. In particular, we have a bilinear composition map
\[ \mathrm{Z}_{n}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) ) \times \mathrm{Z}_ m( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) \rightarrow \mathrm{Z}_{m+n}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) ). \]