Direct image is a Fourier-Mukai transform

Question

I want to show that for a morphism $f: X\to Y$ of smooth projective varieties, say, over $\mathbb{C}$, the (derived) direct image $f_*: D^b(X)\to D^b(Y)$ is a Fourier-Mukai transform. Here, $D^b(X)$ denoted the derived category of bounded coherent sheaves on $X$. So, to show that it is a transform, I need to find a kernel, it is probably $\mathcal{O}_{\Gamma_f}$, where $\Gamma_f\subset X\times Y$ is the graph of $f$. But I am not sure how to actually prove this. We have for $E\in D^b(X)$ $$ \Phi_{\mathcal{O}_{\Gamma_f}}(E)=\pi_{Y*}(\pi_X^* E\otimes \mathcal{O}_{\Gamma_f}) $$ Now I probably need to use the (derived) projection formula $$ f_*(A)\otimes B\simeq f_*(A\otimes f^*(B)) $$ but before that, I probably want to write $\mathcal{O}_{\Gamma_f}$ as an image of the structure sheaf $\mathcal{O}_X$ under some morphism, maybe derived version of the canonical embedding $i: X\to \Gamma_f$?