$\newcommand\sSets{\text{sSets}}\newcommand\sing{\text{sing}}\DeclareMathOperator\Hom{Hom}\newcommand\Top{\text{Top}}$I am looking for a reference describing the notions of homotopy and contractibility in terms of endofunctors of a simplicial category induced by endofunctors of $\Delta$.

Is there a definition of contractibilty in terms of the endofunctor of a simplicial category induced by the décalage endomorphism of $[+1]:\Delta\to \Delta $ shifting dimension by adding a new "always fixed" minimal element to a linear order $$n\mapsto n+1, \ \ f:n\to m \longmapsto f':n+1\to m+1, f'(0):=0; f'(i+1):=f(i) \forall 0\leq i\leq n. $$

The following naïve considerations suggest that a map of sufficiently nice topological space is contractible iff the corresponding map of singular complexes factors through the image with shifted dimension, see the claim below.

Instead of the endomorphism $[+1]:\Delta\to\Delta$ one might want to use $[\times 2]:\Delta\to\Delta$ "doubling" each linear order; $$0\le 1\le \dotsb \le n \longmapsto 0\le 1 \le \dotsb \le n \le 0' \le 1'\le \dotsb\le n'$$ $$n\mapsto 2n+1, \ \ f:n\to m \,\longmapsto\, f':2n+1\to 2m+1, f'(i+n+1)=f'(i)=f(i)\forall 0\leq i\leq n.$$

Claim. Let $F$ and $X$ denote “nice” topological spaces. Further, assume that $F$ is connected (thanks to John Rognes for this correction).

A map $h_0:F\to X$ is contractible, i.e. it factors through the cone of $F$ as $F\xrightarrow{x\mapsto (x,0)} F\times [0,1]/F\times \{1\} \xrightarrow h X$, iff in $\sSets$ the map $\sing F_\bullet \to \sing X_\bullet$ of singular complexes factors via $\sing X_\bullet\circ[+1]$, i.e. the map $\sing F_\bullet \to \sing X_\bullet$ factors as $$\sing F_\bullet \to \sing X_\bullet\circ[+1]\to \sing X_\bullet.$$ Here $\sing X_\bullet\circ[+1]\to \sing X_\bullet$ is the expected map “forgetting the first coordinate”.

Recall that the singular complex is defined using simplices \begin{gather*} \sing F_\bullet(n):=\Hom_{\Top}( \Delta^n, F) \\ \sing X_\bullet(n):=\Hom_{\Top}( \Delta^n, X) \\ \sing X_\bullet\circ[+1](n)=\Hom_{\Top}( \Delta^n\times [0,1]/{\Delta^n\times\{1\}}, X) \end{gather*} where $n\ge 0$, $n\in \Delta$ is the linear order with $n+1$ elements, and $ \Delta^n\times [0,1]/{\Delta^n\times\{1\}}$ is the cone of $n$-simplex $\Delta^n$.

To define a lifting $h_\bullet$, take each map $\delta: \Delta^n \to F$ in $F_\bullet(n)$ to a map $$ h_*(\delta):\Delta^n\times [0,1]/\Delta^n\times\{1\}\to X $$

in $X_\bullet(n+1)$ defined by $$h_*(\delta)( x,t ):= h(\delta(x),t).$$

To see the other direction, note that $h_\bullet:F_\bullet\to X_\bullet[+1]$ takes a singular simplex $\delta:\Delta^n\to F$ into $h_\bullet(\delta):\Delta^{n+1}=\Delta^n\times [0,1]/\Delta^n\times\{1\}\to X$ such that $\delta\circ h_0=h_\bullet(\delta)_{|\Delta^n\times \{0\}}$, i.e. each $\delta:\Delta^n\to F\to X$ factors through the cone of $\Delta^n$.
A verification using functoriality shows that the same factorisation holds for $\mathbb S^n = \partial \Delta^{n+1}$, which means exactly that $h_0$ is weakly contractible, and for “nice” topological spaces contractible and weakly contractible are equivalent.

Alternatively, use that $X_\bullet\circ [+1]$ is simplicially equivalent to the constant simplicial object $n\mapsto X$, $n\geq 0$, by Lemma 1.5.1 of (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.)

Question. Is it true (under some ''niceness'' assumptions) that two maps $f,g:F\to X$ are homotopic iff both maps $sing F_\bullet\xrightarrow {f_\bullet} sing X_\bullet$ and $sing F_\bullet\xrightarrow {g_\bullet} sing X_\bullet$ factor via $X_\bullet[\times 2]$ as $$sing F_\bullet\xrightarrow {\tilde f_\bullet} sing X_\bullet\circ[\times 2] \xrightarrow{pr_{\leq 1/2} } \to sing X_\bullet$$ $$sing F_\bullet\xrightarrow {\tilde g_\bullet} sing X_\bullet\circ[\times 2] \xrightarrow{pr_{\geq 1/2} } \to sing X_\bullet$$ where, as notation suggests, $pr_{\leq 1/2}:X_\bullet\circ[\times 2]\to X_\bullet$ and $pr_{\geq 1/2}:X_\bullet\circ[\times 2]\to X_\bullet$ are maps induced by ''forgetting'' the first/second half of the ''doubled'' linear order $0\le 1 \le ... \le n \le 0' \le 1'\le ...\le n'$.

I am also looking for a reference to these particular claim and question (if it is true).