Left derived functors relative to supplied data are additive functors

By [L1] and [L2], the assignments $A\mapsto L_n^{P}F(A)$ and $u\mapsto L_n^{P}F(u)$ already form a functor.

Let $u,v:A\to B$. Choose comparison lifts ${\tilde{u}}_{\bullet }$ and ${\tilde{v}}_{\bullet }$. By [L3], their degreewise sum ${\tilde{u}}_{\bullet }+{\tilde{v}}_{\bullet }$ is again a chain map, and it lifts $u+v$ because augmentations are additive. Thus it is a comparison lift of $u+v$.

By definition of the derived map and [L5], $L_n^{P}F(u+v)=H_n\text{\hspace{0.17em}⁣}(F({\tilde{u}}_{\bullet }+{\tilde{v}}_{\bullet }))=H_n\text{\hspace{0.17em}⁣}(F({\tilde{u}}_{\bullet }))+H_n\text{\hspace{0.17em}⁣}(F({\tilde{v}}_{\bullet }))\mathrm{.}$ If a different comparison lift of $u+v$ were chosen, [L4] and [L6] would give the same homology map. Hence $L_n^{P}F(u+v)=L_n^{P}F(u)+L_n^{P}F(v)\mathrm{.}$

Step 1.1 gives functoriality, and step 2.1 gives additivity on each hom-group. Therefore [L7] identifies $L_n^{P}F$ as an additive functor.