TITLE: Reference request for statement on nlab: Reedy (co)fibrancy of (co)simplicial objects
QUESTION [5 upvotes]: On the nlab
http://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 it 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.

REPLY [3 votes]: The claim would be true if it were the case that for any simplicial object $X_\bullet$ in some (cocomplete) category $C$, the maps $L_nX\to X_n$ from the latching objects are always monomorphisms.  This is the case when $C$ is any topos for instance.  It is also true when $C$ is any additive category (because of the Dold-Kan theorem on (co)simplicial objects in an additive category with retracts), which would seem to include Faelivrin's example.
But it is not true generally. For instance, if $C$ is the category of augmented commutative $k$-algebras.  Pick an augmented commutative $k$-algebra $A$, and form the simplicial object with $X_n= A\times_k\cdots \times_k A$ ($n$ copies of $A$); this is an example of a 1-coskeleton.  The map $L_2X\to X_2$ has the form
$$
A\otimes_k A \to A\times_k A,
$$
and this can't generally be a monomorphism, e.g., when $A=k[x]$.  
Probably an actually counterexample to the claim can be constructed from an idea like this.