## Definition Let $\dagger$ be a monad on a topos $H$. A morphism $f:X\to Y$ is called *$\dagger$-closed* if $$\array{ X&\to &\dagger X \\ \downarrow^f &&\downarrow^{\dagger f} \\ Y&\to& \dagger Y }$$ is a pullback square. ## Closure properties The class of $\dagger$-closed morphisms satisfies the following closure properties: (1) Every equivalence is $\dagger$-closed. (2) The composite of two $\dagger$-closed morphisms is $\dagger$-closed. (3) The left cancellation property is satisfied: If $h=g\circ f$ and $h$ and $g$ are $\dagger$-closed, then so is $f$. (4) Any retract of a $\dagger$-closed morphism is $\dagger$-closed. (5) The class is closed under pullbacks which are preserved by $\dagger$.