I'm learning homological algebra using several references books. But I find two definitions of derived functor in general triangulated category. I wonder to know which definition is more generally accepted.
Let $\mathcal{D}$ be a triangulated category and $\mathcal{C}$ a strictly full triangulated subcategory of $\mathcal{D}$ with the quotient $Q:\mathcal{D}\to \mathcal{D}/\mathcal{C}$. Given a triangulated category $\mathcal{T}$ and a triangulated functor $F:\mathcal{D}\to \mathcal{T}$, I have seen two different definitions of derived functor of $F$ with respect to $\mathcal{C}$.
Version 1 A right derived functor of $F$ with respect to $\mathcal{C}$ is a pair consisting of a triangulated functor $\mathcal{R}F:\mathcal{D}/\mathcal{C}\to \mathcal{T}$ and a trinatural transformation $\eta:F\Rightarrow (\mathcal{R}F)Q$ such that: given any triangulated functor $G:\mathcal{D}/\mathcal{C}\to \mathcal{T}$ and trinatural transformation $\xi:F\Rightarrow G Q$, there is a unique trinatural transformation $\chi:\mathcal{R}F\Rightarrow G$ such that $\xi=(\chi Q)\eta$.
For example, this definition occurs in note1 (page 198) and note2 (page 72).
Version 2 A right derived functor of $F$ with respect to $\mathcal{C}$ is just a left Kan extension $(\text{Lan}_QF,\eta)$. Also, some references require $\text{Lan}_QF$ to be triangulated without $\eta$ being compatible with translations.
For example, this definition occurs in page 253 of "Categories and Sheaves" by Masaki Kashiwara (eseential the same) and some books in other languages.
I think these two definition are not equivalent. A ordinary Kan extenstion $\text{Lan}_QF$ has no reason to be triangulated, and its natural transformation $\eta$ also has no reason to be compatible with translations. On the converse, the left Kan extension in the sense of triangulated categories (version 1) can not be an ordinary left Kan extension.
I wonder to know which definition is more generally accepted.