Assumptions: filtered
Conclusions: sifted
Reason: Every filtered category C\mathcal CC is inhabited and has final diagonal functors Δ:C→CJ\Delta:\mathcal C\to\mathcal C^JΔ:C→CJ for all finite index categories JJJ; in particular, it is inhabited and its diagonal Δ:C→C×C\Delta:\mathcal C\to\mathcal C\times\mathcal CΔ:C→C×C is final.