# Cohomology functors in a triangulated category with t-structure.

Given a triangulated category $$\mathcal{D}$$ with a t-structure $$(\mathcal{D}_{\leq 0}, \mathcal{D}_{\geq 0})$$ the cohomology functor can be defined as $$$$H^k := \tau_{\geq 0}\circ \tau_{\leq 0} \circ [k] : \mathcal{D} \rightarrow \mathcal{A}$$$$ where $$\mathcal{A} := \mathcal{D}_{\leq 0} \cap \mathcal{D}_{\geq 0}$$ is the heart of the t-structure. It is known that (e.g. https://en.wikipedia.org/wiki/T-structure#Cohomology_functors), given an exact triangle $$A \rightarrow B \rightarrow C \rightarrow A[1]$$ we get the long-exact-sequence $$$$\cdots \rightarrow H^k(A) \rightarrow H^k(B) \rightarrow H^k(C) \rightarrow H^{k+1}(A) \rightarrow \cdots$$$$ just like in the usual cohomological algebra. In the case of derived categories, I could see why this should be true (mainly because the cohomology functor simply computes the usual cohomology).

But is it easy to see that we get a long-exact-sequence from an exact triangle using only the definition of t-structures and the axioms of the general triangulated categories? Could someone suggest to me the easy way to do this?