A morphism A→ϕBA \xrightarrow{\phi} BAϕB is point-surjective iff for every point 1→qB1 \xrightarrow{q} B1qB, there exists a point 1→pA1 \xrightarrow{p} A1pA that lifts qqq (satisfy ϕ∘p=q\phi \circ p = qϕ∘p=q).