Ext is hom in the derived category

In the projective case take $P\to M[0]$ with $P^i=0$ for $i>0$. By [F3], $P$ is K-projective. Replacing the source and removing roofs gives ${\mathrm{Hom}}_D(M,N[n])={\mathrm{Hom}}_K(P,N[n])$. Reindexing the degree-zero Hom theorem gives the latter as $H^n\underset{}{\mathrm{Hom}}(P,N)$. This applies also to $M=0$ or $N=0$.

In the injective case take $N[0]\to I$ with $I^i=0$ below zero. By [F4], $I$ and each shift $I[n]$ are K-injective. Replacing the target and removing roofs gives ${\mathrm{Hom}}_D(M,N[n])={\mathrm{Hom}}_K(M,I[n])=H^n\underset{}{\mathrm{Hom}}(M,I)$. Here the differential is exactly the classical injective Ext differential, so no correction is needed.

The classical projective Ext differential is precomposition with the resolution differential, whereas the cochain Hom differential on degree $q$ maps into $N[0]$ is $(-1{)}^{q+1}$ times precomposition. Multiplication in degree $q$ by $(-1{)}^{q(q+1)\mathrm{/}2}$ is a complex isomorphism from the classical complex to this Hom complex. It identifies their cohomology in all $n\ge 0$, with sign $+1$ in degree zero.

For a morphism of objects, the corresponding comparison between their supplied resolutions is the unique homotopy class representing that morphism after localization: existence and uniqueness follow from [F1] on the projective side and [F2] on the injective side. Consequently the Hom identifications are natural in both objects. When both resolutions exist, put $T=\underset{}{\mathrm{Hom}}(P,I)$. The maps $\underset{}{\mathrm{Hom}}(P,N)\to T\leftarrow \underset{}{\mathrm{Hom}}(M,I)$ induce cohomology isomorphisms: by [F5], each degree-$n$ map is the map on homotopy Hom, which [F1] or [F2] identifies with composition by the corresponding invertible resolution map in $D$. Composing this span with the classical-projective sign isomorphism of step 2.1 defines the mixed-complex identification of the two classical Ext constructions. Their maps to derived Hom agree under this identification, since the two localized composites agree. In degree zero every sign is $+1$ and ordinary morphisms, including identities, are preserved.