This is Cisinski & Deglise's Triangulated categories of mixed motives. In this section. $\mathscr P$ is assumed be smooth morphisms. As the picture shows, $f,g$ are smooth separaed morphism of finite type over $S$, where $S$ is noetherian. And $s,t$ are sections of $f,g$ respectly.
In the lemma 2.4.6 below, authors state use $\mathscr P$-base change formula. I don't how can I use the $\mathscr P$-base change formula. It seems that $s$ is a smooth morphism. Is this true? Anyone could help me?
The section $s : S \to X$ is generally not smooth. In fact, since $f :X \to S$ is separated, we have that $s$ is smooth if and only if it is a clopen immersion.
Okay, so let's prove $Ex(\Delta_{\sharp *}) : p_\sharp s'_* \to s_* p'_\sharp$ is an equivalence. Let's write $j : U = X\smallsetminus s(S) \to X$ for the open complement of $s$.
Since $\mathscr{T}$ satisfies $(Loc_s)$, it suffices to show that $s^* Ex(\Delta_{\sharp *}) : s^* p_\sharp s'_* \to s^* s_* p'_\sharp $ and $j^* Ex(\Delta_{\sharp *}) : j^* p_\sharp s'_* \to j^* s_* p'_\sharp$ are equivalences.
For $j^* Ex(\Delta_{\sharp *})$, we can use $\mathscr{P}$-basechange to show the source and target are $(p_U)_\sharp j'^* s'_* \to j^* s_* p'_\sharp$. (Here, $p_U : Y\times_X U \to U$ and $j' : Y\times_X U \to Y$ are coming from the pullback of $p$ and $j$.) Note that $j'$ is the open complement of $s'$. Using the relations from Section 2.3.1, both the source and the target are trivial.
I will leave the case of $s^* Ex(\Delta_{\sharp *})$ to you.