This suggests that by preceding the adjoint in question by the embedding $I$ we shall obtain the “existential quantification” (which sends $X$ to $\Omega\Sup{X}$ and $f$ to $\exists\Sub{f}$) from $\ECat$ to $\ECat$. Indeed, the image of the fibered product of $\in\Sub{X}$ and $\prn{\Idn{X},f}$ — therefore of $\in\Sub{X}$ — over $\Omega\Sup{X}\times Y$ is indeed the vertical composite depicted below: