The matrix-factorization Hom differential is a dg differential

If $\varphi$ has parity $p$, then $D\varphi$ has parity $p+1$. Expanding once more gives $D^2\varphi ={\delta }_F^2\varphi -(-1{)}^p{\delta }_F\varphi {\delta }_E-(-1{)}^{p+1}{\delta }_F\varphi {\delta }_E-\varphi {\delta }_E^2\mathrm{.}$ The middle coefficients sum to zero, while [F1] turns the first and last terms into $W\varphi -\varphi W=0$, since $W$ acts centrally as a scalar. Thus $D^2=0$.

For homogeneous $\psi :F\to G$ of parity $q$, expand both sides: $D(\psi \varphi )={\delta }_G\psi \varphi -(-1{)}^{p+q}\psi \varphi {\delta }_E$, whereas $D\psi \varphi +(-1{)}^q\psi D\varphi$ equals the same expression because the two intermediate terms $(-1{)}^q\psi {\delta }_F\varphi$ cancel. This is the graded Leibniz rule.

Composition is associative because it is composition of sheaf maps, and for each $E$, $D({\mathrm{id}}_E)={\delta }_E-{\delta }_E=0$. Together with [step 1.1], the morphism differential squares to zero and preserves the identities, proving the dg-category axioms.