The cancellation theorem for properties of morphisms

A base-change-stable compositional property cancels through a morphism once that morphism's diagonal has the property.Suggest a slogan

(Cancellation)

Let $P$ be a class of morphisms of schemes stable under base change and composition (D-MORLOCAL). Let $\pi \colon X \to Y$ and $\rho \colon Y \to Z$ be morphisms and $\tau = \rho \circ \pi$. If $\tau \in P$ and the diagonal $\delta_\rho \colon Y \to Y \times_Z Y$ is in $P$, then $\pi \in P$.

Let $P$ be stable under base change and composition, and $\rho \circ \pi \in P$.

If $P$ contains all locally closed immersions, then $\pi \in P$.
If $P$ contains all closed immersions and $\rho$ is separated, then $\pi \in P$ (P-AGH248PROPSTAB, part (e)).
If $P$ contains all quasicompact morphisms and $\rho$ is quasiseparated, then $\pi \in P$.

Proper morphisms are stable under base change and composition and contain the closed immersions. If $X$ and $Y$ are proper over a field $k$, with structure morphisms $\tau \colon X \to \Spec k$ and $\rho \colon Y \to \Spec k$, then every $k$-morphism $\pi \colon X \to Y$ is proper, by part 2 of the corollary, since $\rho$ is separated.

If $\rho \circ \pi$ has a property $P$ stable under base change and composition, when does $\pi$ have $P$?

Why is a morphism between proper $k$-schemes proper?

Dependencies

D-MORLOCALClasses of morphisms stable under base change and local on the source or targetuses
D-T2J3QSeparated and quasi-separated morphismsuses

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Classes of morphism