Implication Details

Assumptions: exact filtered colimits

Conclusions: filtered-colimit-stable monomorphisms

Reason: This is because f:X⟶Yf : X \longrightarrow Y is a monomorphism iff the diagram X→idXid↓↓fX→fY\begin{CD} X @>{\mathrm{id}}>> X \\ @V{\mathrm{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.