in which the map \(A \hookrightarrow B\) is a monomorphism. Then also \(C \to D\) is a monomorphism and the square is a pullback square.
Proof
Consider the cube The top and bottom faces are pushouts, and the left and back faces are pullbacks. By descent (Mather's first cube lemma), the front and right faces are also pullback squares. This says precisely that the starting square is a pullback square and that the map \(C \to D\) is a monomorphism.
References
Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. A generalized Blakers-Massey theorem. J. Topol., 13 (4), 1521–1553. 2020.
