# Transferring $t$-structures via adjoint functors

In Gaitsgory and Rozenblyum's Derived Algebraic Geometry book, they frequently use the following technique to transfer a $t$-structure from one category to another (for example, 1.5 in this paper or page 58 of this paper). The claim is (I think -- maybe some hidden assumptions?):

Let $F: C \rightarrow D$ be an exact functor which is a left adjoint, and suppose $D$ has a $t$-structure. Then, we define $C^{\leq 0}$ to be the full subcategory whose objects satisfy $F(X) \in D^{\leq 0}$. We define $C^{\geq 1}$ to be the right orthogonal to $C^{\leq 0}$. This defines a $t$-structure.

For example, in the first link, a proposition in Lurie's Higher Algebra is referenced. But this proposition assumes that the functor in question is a localization functor, which is exactly what I'm not sure about.

Here's an attempt. Let $Y \in C$. Then, there is an exact triangle $X \rightarrow FY \rightarrow Z$ where $X \in D^{\leq 0}$ and $Z \in D^{\geq 1}$. We have a map $Y \rightarrow GZ$ where $G$ is the right adjoint -- one can check that the right adjoint takes $D^{\geq 1}$ to $C^{\geq 1}$. The claim is that the fiber (i.e. cocone) of $Y \rightarrow GZ$ is in $C^{\leq 0}$.

Now, the only way to check this is to apply $F$. Since $F$ is exact, we have $$F(cocone) \rightarrow FX \rightarrow FGZ$$ However, it's not clear that the cocone of $FX \rightarrow FGZ$ is in $D^{\leq 0}$ to me.