Equalizer diagram equipped with other arrows

I am trying to draw this diagram and can't get slopped arrows right.

\documentclass{article}
$\begin{tikzcd}[row sep=1.4cm,column sep=1.6cm] & FU_i \arrow[r,"F(U_i\cap U_j\subset U_i)"] & F(U_i\cap U_j)\\ FU \arrow[r,dashed,"e"] \arrow[dr] \arrow[ur] & \prod\limits_i FU_i \arrow[d] \arrow[u] \arrow[r,dashed,"q"',yshift=-2pt] \arrow[r,dashed,"p",yshift=2pt]& \prod\limits_{i,j}F(U_i\cap U_j) \arrow[d] \arrow[u]\\ & FU_j \arrow[r,"F(U_i\cap U_j\subset U_j)"']& F(U_i\cap U_j) \end{tikzcd}$