Let $X,Y$ be smooth varieties defined over $k$. Suppose $P$ is a coherent sheaf on $X \times Y$ flat over $X$, considering the Fourier-Mukai transform $$\Phi_P : D^{b}(X) \to D^{b}(Y)$$ which is defined as $$F \mapsto q_* (P \otimes p^{*}F).$$

Suppose $k(x)$ is the skyscraper sheaf on the closed point $x \in X$, and $k(x) \cong k$. Then how to justify the following claim: $$\Phi(k(x)) \cong P_x$$ where $P_x := P \mid_{ \{ x \} \times Y}$ is considered as a sheaf on $Y$ via the second projection: $\{x\} \times Y \to Y$ ?

my difficulties comes at two places: (1)All the definition here are in the derived sense, so I have no idea of compute them efficiently. (2) It seems $P \mid_{ \{ x \} \times Y}$ is a pullback rather than pushforward which is what the claim looks like.

Moreover, where is the condition $P$ is flat used in the calculation?