I am new to MO and I hope that this question is suitable for it. Following Hovey's "Model Categories" ( http://ericmalm.net/ac/projects/symmetric-spectra/hovey--model-cats.pdf ) to study model structures on functor categories, it seems that something is missing, or better I proved something to proceed further, but I am not really sure about this result.
It seems to me that, considering a model category $\mathscr{C}$ and the simplicial category $\Delta$ and endowing $\mathscr{C}^{\Delta}$ with the Reedy model structure, there is a natural isomorphism $L_n \ l^{\bullet}A \simeq l^{\bullet}A [n] \ \ \forall A \in \mathscr{C}$.
Let me explain a bit the notation (but everything can be found on pages 124 to 127 in the previous link): by $L_n$ is meant the $n$-th latching space functor ( http://ncatlab.org/nlab/show/Reedy+model+structure) and $l^{\bullet}A$ is the cosimplicial object in $\mathscr{C}$ whose $n$-th "space" is the $n+1$-fold coproduct of $A$, with the obvious cosimplicial structure. Furthermore, it is left adjoint to the evaluation functor $Ev_0 : \mathscr{C}^{\Delta} \to \mathscr{C}$
The abovementioned isomorphism comes out by unraveling definitions and using universal properties of colimits and adjointness, at least for what I have tried.
Could someone confirm this fact, or disprove it (so that I have to find another way to prove the claim I am interested into)?
Thanks in advance, any help will be highly appreciated.