I would have probably very short and simple question. In https://ncatlab.org/nlab/show/%C4%8Cech+nerve there is a statement
Čech nerve of a morphism $f$ is the internal nerve of the internal groupoid corresponding to the kernel pair of this morphism.
What is actual multiplication in this groupoid? Or more generally, how to find all faces and degeneracies of this simplicial object? I tried to see all these maps only from morphisms which create limiting cones of $k$-folded pullbacks, but I failed to see all of them. If this is the right way how to obtain all faces and degeneracies I would ask for a brief explanation of how to obtain face map corresponding to the groupoid multiplication in question.