Suppose I have triangulated categories $A \xrightarrow{i_{*}} B \xrightarrow{j^{*}} C$ where $i_{*},j^{*}$ have right adjoints $i^{*}, j_{*}$ respectively (they possibly also have left adjoints). Suppose further that for every $X \in B$ there is an exact triangle
$$i_{*}i^{*}X \to X \to j_{*}j^{*}X \to i_{*}i^{*}X[1]$$
I want to say that I can take two t-structures $A^{\leq 0}, C^{\leq 0}$ on $A$ and $C$ and glue them to t-structure on $B$. Can this actually be done?
My attempt was to take $B^{\leq 0}$ to be the full subcategory of objects $X: i^{*}X \in A^{\leq 0}, j^{*}X \in C^{\leq 0}$. However I can't seem to construct the required adjunction because of the non-functoriality of the cone. Is there an obvious way round this issue?
All this is written in detail in Asterisque 100 (A.A. Beilinson, J. Bernstein, Pierre Deligne, Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Aste’risque 100, Soc. Math. France, Paris 1982).