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.