Let $T$ be a triangulated category closed with respect to (small) coproducts, and $t$ be (an arbitrary!) a $t$-structure on $T$. I have noted that the heart $\underline{Ht}$ of $t$ is closed with with respect to coproducts as well (that it, it is an AB3 abelian category).


Indeed, the aisle $T^{t\le 0}$ is closed with with respect to coproducts. Hence for $H\_i\in \underline{Ht}$ the zeroth $t$-homology of the $T$-coproduct of $M\_i$ gives the $\underline{Ht}$-coproduct of these objects (by the corresponding adjunction).


Is this correct? Did anybody formulate this (rather counter-intuitive) statement earlier?



Are hearts of all $t$-structures on smashing triangulated categories closed with respect to coproducts (also)?
This is Proposition 3.1.2 in the [thesis of Parra](https://arxiv.org/abs/1409.6639), "Hearts of $t $-structures which are Grothendieck or module categories".


