On the nlab

ncatlab.org/nlab/show/Reedy+model+structure#fibrant_and_cofibrant_objects

it is claimed that a simplicial object in a model category, in which all monomorphisms are cofibrations, is always Reedy-cofibrant.

Does anyone know a reference for this?

In the case of simplicial sets the statement is Theorem 15.8.7 in Hirschhorn's book.

Follow up question: Is is dually true that any cosimplicial object in a model category, in which all epimorphisms are fibrations, is Reedy-fibrant?

Again, is there a reference?

Thanks a lot.