for general $E$, i'll write
$$ ( E_{hC_p} \to E^{C_p} \to E^{\Phi C_p} ) \to ( E_{hC_p} \to E^{hC_p} \to E^{tC_p}) . $$
for $E = C_p/e$, i believe this is:
$$ (S \to S \to 0 ) \to ( S \to (C_p/e)^{hC_p} \to (C_p/e)^{tC_p} ).$$
(i don't know how to identify the last two terms.) for $E = C_p/C_p$, i believe this is:
$$ ( S_{hC_p} \to S_{hC_p} \oplus S \to S ) \to (S_{hC_p} \to S^{hC_p} \to S^\wedge_p ). ) $$
of course, i can also identify $S_{hC_p} \simeq \Sigma^\infty_+ BC_p$. i don't know how to identify $S^{hC_p}$ otherwisely, either; equivalently, i don't know what the gluing map $S^\wed…