Let $[+1]:\Delta\to\Delta$ be the decalage endomorphism sending $n\mapsto n+1$
adding a new minimal element, i.e. $f:n\to m$ is sent to $f':n+1\to m+1$, $f'(0)=0$, and for $0\leq i \leq n$
$f'(i+1)=f(i)+1$.

For $Y\in \operatorname{Ob} sSets$, let $Y^{\Delta[1]}$ denote the internal hom of maps of $\Delta[1]$ to $Y$. 

Finally, let $const(Y_0)$ denote the discrete object $n\mapsto Y_0$.
Recall that taking the 0-face defines a map $Y\circ[+1]\to const(Y_0)$ and it is a weak equivalence
by  (Lemma 1.5.1, K.Waldhausen, Waldhausen, Friedhelm Algebraic K-theory of spaces. Algebraic and geometric topology (New Brunswick, N.J., 1983), 318-419, Lecture Notes in Math., 1126, Springer, Berlin, 1985.) (thanks to @John Rognes in https://mathoverflow.net/questions/428423/defining-homotopy-via-endofunctors-of-a-simplicial-category ).


Is it true that 

> $Y\circ [+1]$ is isomorphic to $const(Y_0)\times_Y Y^{\Delta[1]}$ ? 

Here $const(Y_0)\to Y$ is the obvious diagonal map, and $Y^{\Delta[1]\to Y$ is one of the two standard projections. 


Sketch of possible proof: Let us construct a map  $Y\circ [+1] \to Y^{\Delta[1]}$
which induced the isomorphism together with the map $Y\circ[+1]\to const(Y_0)$.

Let $n$ denote the linear order $[0<1<...<n]$.

A map $(Y\circ [+1])_n=Y_{n+1}=Hom(\Delta[n+1],Y) \to (Y^{\Delta[1]})_n=Hom(\Delta[n]\times \Delta[1],Y)$
is given by a map $\Delta[n]\times \Delta[1]\to \Delta[n+1]$. 
To define this map, let us define for each $N$ a map
$Hom(N,n)\times Hom(n,1) \to Hom(N,n+1)$
by $(f,g)\mapsto (f+1)g$. 

Note that when $g:N\to 1 $ sends everything to $0\in[0<1]$,
the map $(f+1)g:N\to n+1$ sends everything to $0\in[0<1<...<n+1]$, and therefore $Y\circ[+1]\to  Y^{\Delta[1]}\xrightarrow{0\in[0<1]} Y$ is the map $Y\circ[+1]\to const(Y_0)\to Y$.


Note that when $g:N\to 1 $ sends everything to $1\in[0<1]$,
the map $(f+1)g:N\to n+1$ sends each $i$ to $i+1$, and therefore $Y\circ[+1]\to  Y^{\Delta[1]}\xrightarrow{1\in[0<1]} Y$ is the map $Y\circ[+1]\to  Y$.

Does this argument really work ? Is there a reference for the fact it aims to prove ? 


The motivation is that if the fact is true, then the property 
> a map $X\to Y$ factors though $Y\circ[+1]\to Y$

defines the class of maps homotopic to a map $X\to const(Y_0)$,
i.e.  homotopically trivial on each connected compoment of the domain. 
And this property can be formulated in an arbitrary category of simplicial objects
defining a class of almost homopically trivial maps. 
I am also looking for references studying this property for 
an arbitrary category of simplicial objects.