CatDat

Implication Details

Claim: If a category has exact filtered colimits, then it has filtered-colimit-stable monomorphisms.

Proof: This is because $f : X \longrightarrow Y$ is a monomorphism iff the diagram $$\begin{CD} X @>{\id}>> X \\ @V{\id}VV @VV{f}V \\ X @>>{f}> Y \end{CD}$$ is a pullback, and if a functor preserves finite limits, it preserves pullbacks in particular.