Let $f: X \to Y$ be a map of CW-complexes with continuous maps as morphisms.


How would one show that homotopy pullback $\mathcal D/Y → \mathcal D/X$ is right adjoint?


Here $\mathcal D$ is the derived category (edit, I mean the category of CW-complexes with continuous maps).



Homotopy pullback is right adjoint in the derived category
There is no such functor $\mathcal D/Y\to \mathcal D/X$. It's clear what is meant to be on objects, but it is not well-defined on morphisms.


Let $f$ be the inclusion of a point $p$ into a circle $C$. Let $g$ and $h$ be the inclusions of two points, say $q$ and $r$, into $C$. Regard $g$ and $h$ as objects of $\mathcal D/C$. There is a unique isomorphism between them. The homotopy pullback of $g$ along $f$ is the space of paths from $p$ to $q$ in $C$. The homotopy pullback of $h$ along $f$ is the space of paths from $p$ to $r$ in $C$. What map is the isomorphism between $g$ and $h$ meant to give from the one homotopy pullback to the other?


