t-cohomology is functorial, additive, shift-compatible and detects amplitude

Claim (i). The formula $H_t^n(X)=({\tau }_{\le n}{\tau }_{\ge n}X)[n]$ is a composite of functors by [F1] and [F2], and the induced maps are uniquely compatible with the truncation maps by [F1]. Put $Y={\tau }_{\ge n}X$ and $Z={\tau }_{\le n}Y$. For $T\in {\mathcal{T}}^{\le n-1}$, adjunction gives $\mathrm{Hom}(T,Z)\cong \mathrm{Hom}(T,Y)=0$, since $Y\in {\mathcal{T}}^{\ge n}$. The aisle criterion therefore puts $Z$ in ${\mathcal{T}}^{\ge n}$; it is in ${\mathcal{T}}^{\le n}$ by construction. Consequently $Z[n]=H_t^n(X)$ lies in the heart by the shift conventions.

Claim (ii). For all $a,b$ there is a canonical isomorphism ${\tau }_{\le a}(X[b])\cong ({\tau }_{\le a+b}X)[b]$ over $X[b]$, and dually for ${\tau }_{\ge a}$: for $T\in {\mathcal{T}}^{\le a}$ both objects represent the same functor, $\mathrm{Hom}(T,{\tau }_{\le a}(X[b]))\cong \mathrm{Hom}(T,X[b])$ by the adjunction [F1], and $\mathrm{Hom}(T,({\tau }_{\le a+b}X)[b])\cong \mathrm{Hom}(T[-b],{\tau }_{\le a+b}X)\cong \mathrm{Hom}(T[-b],X)\cong \mathrm{Hom}(T,X[b])$ because $T[-b]\in {\mathcal{T}}^{\le a+b}$, so the essential uniqueness in [F1] identifies them. Hence $H_t^n(X[m])=({\tau }_{\le n}{\tau }_{\ge n}(X[m]))[n]\cong ({\tau }_{\le n}(({\tau }_{\ge n+m}X)[m]))[n]\cong (({\tau }_{\le n+m}{\tau }_{\ge n+m}X)[m])[n]\cong H_t^{n+m}(X)$, using the coherence isomorphisms of [F2], and the isomorphisms are natural in $X$ by functoriality of the truncations.

Claim (iv), the forward implications. If $X\in {\mathcal{T}}^{\le m}$ and $n>m$, then $X\in {\mathcal{T}}^{\le n-1}$ by [F1], so ${\tau }_{\ge n}X=0$ by the aisle criterion [F1] and $H_t^n(X)=0$. If $X\in {\mathcal{T}}^{\ge m}$ and $n<m$, then $X\in {\mathcal{T}}^{\ge n+1}$ by [F1], so ${\tau }_{\ge n}X\cong X$ and ${\tau }_{\le n}{\tau }_{\ge n}X={\tau }_{\le n}X=0$ by the dual aisle criterion, whence $H_t^n(X)=0$.

Claim (v), concentration. The identity $H_t^m(X)[-m]={\tau }_{\le m}{\tau }_{\ge m}X$ is the definition. If $X\in {\mathcal{T}}^{\ge m}$ then ${\tau }_{\ge m}X\cong X$ by [F1], giving the second identity. If $X\in {\mathcal{T}}^{\le m}$, then ${\tau }_{\ge m}X\in {\mathcal{T}}^{\le m}$: for $W\in {\mathcal{T}}^{\ge m+1}$ one has $\mathrm{Hom}({\tau }_{\ge m}X,W)\cong \mathrm{Hom}(X,W)=0$ by the adjunction and orthogonality of [F1], so the aisle criterion gives ${\tau }_{\ge m}X\in {\mathcal{T}}^{\le m}$; hence ${\tau }_{\le m}{\tau }_{\ge m}X\cong {\tau }_{\ge m}X$, giving the first identity. If $X$ lies in both aisles, then ${\tau }_{\le m}X\cong X$ and ${\tau }_{\ge m}X\cong X$, and the composite canonical map $X\cong H_t^m(X)[-m]$ is the truncation map.

Claim (iii). The inclusion ${\mathcal{T}}^{\le n}\hookrightarrow \mathcal{T}$ has the right adjoint ${\tau }_{\le n}$ by [F1], and the inclusion ${\mathcal{T}}^{\ge n}\hookrightarrow \mathcal{T}$ has the left adjoint ${\tau }_{\ge n}$; these inclusions and $\mathcal{T}$ are additive by [F3], so ${\tau }_{\le n}$ and ${\tau }_{\ge n}$ are additive functors by [F3], and so is the autoequivalence $[n]$ by [F2]. Hence $H_t^n$ is additive, and by [F3] it preserves the split biproduct diagram of $X\oplus Y$: the images of the injections and projections under $H_t^n$ are the two mutually inverse structure maps of a biproduct diagram of $H_t^n(X)$ and $H_t^n(Y)$, which is precisely the assertion, and the resulting isomorphism $H_t^n(X\oplus Y)\cong H_t^n(X)\oplus H_t^n(Y)$ is natural by functoriality.

Claim (iv), the converses for a bounded t-structure. Suppose $H_t^n(X)=0$ for all $n>m$. By boundedness choose $N\ge m$ with $X\in {\mathcal{T}}^{\le N}\cap {\mathcal{T}}^{\ge -N}$; then $X\in {\mathcal{T}}^{\le N}$, and if $X\in {\mathcal{T}}^{\le k}$ for some $k\ge m+1$, the concentration identity of step 1.4 gives ${\tau }_{\ge k}X\cong H_t^k(X)[-k]=0$, so $X\cong {\tau }_{\le k-1}X\in {\mathcal{T}}^{\le k-1}$ by the aisle criterion. Descending finitely from $N$ to $m+1$ gives $X\in {\mathcal{T}}^{\le m}$. The lower bound is dual: if $H_t^n(X)=0$ for all $n<m$, choose $N\ge -m$ with $X\in {\mathcal{T}}^{\le N}\cap {\mathcal{T}}^{\ge -N}$ and ascend from $-N$ to $m-1$, using ${\tau }_{\le k}X\cong H_t^k(X)[-k]=0$ to conclude $X\to {\tau }_{\ge k+1}X$ is an isomorphism and $X\in {\mathcal{T}}^{\ge k+1}$; the two arguments are independent.

Claim (v), map detection. Let $X,Y\in {\mathcal{T}}^{\le m}\cap {\mathcal{T}}^{\ge m}$. By step 1.4 the canonical maps give isomorphisms $X\cong H_t^m(X)[-m]$ and $Y\cong H_t^m(Y)[-m]$; under these, a morphism $f:X\to Y$ corresponds to $H_t^m(f)$: both the source and target identifications are the truncation maps, and $H_t^m$ is a functor commuting with the truncations by step 1.1, so the induced bijection is $f\mapsto H_t^m(f)$, and it is a bijection because the identifications are isomorphisms.

Claim (v), the bounded single-degree case: if the t-structure is bounded and $X,Y$ have $H_t^n=0$ for every $n\mathrm{\ne }m$, then step 2.2 (applied to the positive and negative ranges) gives $X,Y\in {\mathcal{T}}^{\le m}\cap {\mathcal{T}}^{\ge m}$, so step 2.3 applies and $\mathrm{Hom}(X,Y)\to \mathrm{Hom}(H_t^m(X),H_t^m(Y))$, $f\mapsto H_t^m(f)$, is a bijection under the canonical identifications.

Steps 1.1 to 1.4 and 2.2 to 3.1 prove all the clauses: (i) in step 1.1, (ii) in step 1.2, (iii) in step 2.1, both directions of (iv) in steps 1.3 and 2.2, and the concentration identities and map detection of (v) in steps 1.4, 2.3 and 3.1.