Classical derived functors are the cohomology objects of the total derived functor

For $M[0]$ choose a resolution supported in degrees $\le 0$ on the projective side or $\ge 0$ on the injective side. Up to the comparison homotopy equivalence with the supplied replacement, the model complex is precisely the image of the deleted classical resolution, after $P_n=P^{-n}$. Its cohomology is therefore exactly the stated classical derived object, including $M=0$ and $n=0$. Comparison maps are the same homotopy classes on both constructions.

A complex with cohomology below $a$ zero admits an injective model zero below $a$; its image under $F$ retains this support. The projective construction gives the dual upper-bound assertion. Apply exact $RF$ to the truncation triangle: the tail $RF({\tau }^{\ge a+1}X)$ has zero cohomology in degrees $\le a$, and its preceding degree is zero too. The long exact sequence gives the stated isomorphism for $i\le a$. The dual argument with the head supported at most $a-1$ gives the $LF$ isomorphism for $i\ge a$.

For a short exact sequence choose projective horseshoes using [F7]. For the injective side apply the projective assertion of [F7] to $0\to C\to B\to A\to 0$ in the abelian category ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}$ of [F8]: the supplied injective resolutions become projective resolutions there. Reversing all arrows gives chain maps $0\to I_A\to J_B\to I_C\to 0$ and a splitting in every degree. Thus both sides have a degreewise split short exact sequence of resolutions, not merely biproduct middle objects. Applying additive $F$ preserves this splitting. Comparison homotopy equivalences identify the auxiliary middle resolution with the supplied one.

Write either image sequence in cochain form $0\to E\stackrel{i}{\to }G\stackrel{q}{\to }H\to 0$. Choose degreewise splittings $s:H\to G$ and $\pi :G\to E$. The off-diagonal map $t=\pi d_{G}s:H^j\to E^{j+1}$ satisfies $d_{G}s-s{d}_H=it$ and $d_{E}t+t{d}_H=0$. It induces the classical cycle-lifting boundary $[t]$. The map $H\to \mathrm{Cone}(i)$ given by $(s,-t)$ is a complex map, and its composite with $e(g,e^{\mathrm{\prime }})=q(g)$ is identity. Since $e$ is the quasi-isomorphism used in [F10], the triangle arrow $p{e}^{-1}$ induces $-[t]$, with $p(g,e^{\mathrm{\prime }})=e^{\mathrm{\prime }}$. Consequently the identity cohomology identifications intertwine boundaries up to minus one. Multiplication in degree $n$ by $(-1{)}^n$ fixes this on both sides: $(-1{)}^{n+1}(-1)=(-1{)}^n$ on the right and $(-1{)}^{n-1}(-1)=(-1{)}^n$ on the left. The formulas are biproduct-morphism identities, valid in any abelian category. Splitting changes give homotopic maps, and the replacement comparisons are natural, so these are natural comparisons of delta functors with identity comparison at degree zero.

On objects the support result and the exact triangle of a short exact sequence give $0\to H^{0}RF(A)\to H^{0}RF(B)\to H^{0}RF(C)$ and $H^{0}LF(A)\to H^{0}LF(B)\to H^{0}LF(C)\to 0$. Hence these degree-zero functors have the stated side-exactness. If $F$ is left exact, $0\to F(M)\to F(I^0)\to F(I^1)$ is exact, identifying $F(M)$ with $H^{0}RF(M)$. Right exactness similarly identifies the cokernel $H^{0}LF(M)$ with $F(M)$. Conversely an isomorphic functor inherits the corresponding exactness.

Under the stated side-exactness the comparison already induces an isomorphism in degree zero, and both complexes have zero cohomology on the opposite side. It is therefore a quasi-isomorphism iff all positive right derived objects (or all positive left derived objects) vanish. This proves both implications of each acyclic-object criterion.

In the right-hand long exact sequence $0\to F(A)\to F(B)\to F(C)\to R^{1}F(A)\to R^{1}F(B)\to R^{1}F(C)\to \cdots$, vanishing for $(A,C)$ gives vanishing for $B$ degree by degree, and vanishing for $(A,B)$ gives vanishing for $C$. For $(B,C)$ all degrees of $A$ above one vanish and $R^{1}F(A)=\mathrm{coker}(F(B)\to F(C))$; this is exactly the extra epic condition. Each case also makes the displayed $F$ sequence short exact. Reversing arrows and reindexing gives the three left cases; the last obstruction is $L_{1}F(C)=\ker (F(A)\to F(B))$.

Finally impose DC and both supplied object-resolution data, exactly as in the published universality theorem, together with the appropriate enough-objects and side-exactness hypothesis. That theorem gives universality of the classical delta functor. The natural object identifications and sign-adjusted connecting comparisons in steps 1.1 and 3.1 transfer it to the cohomology description. No universality assertion with weaker assumptions is inferred from that citation.