I am looking for a reference explicitly defining simplicial homotopy in terms of endofunctors of $\Delta$, and developing homotopy theory in this terms. The following is a particular question.

Is it possible to rephrase the following definition of simplicial homotopy in e.g. (8.3.11 of An introduction to homological algebra by C. Weibel) in terms of endofunctors of $\Delta$ ?

We say $f$ and $g$ are (strictly) simplicial homotopic if if for each $p\geq 0$, there exists morphisms $$ h_i=h^p_i:X_p\to Y_{p+1} \text{ for } i=0,\ldots,p $$ such that the following conditions hold.

$$\partial_0 h_0=f_p, \partial_{p+1}h_p=g_p;$$

$$ \partial_ih_j=\begin{cases}h_{j-1}\partial_i & i<j\\ \partial_ih_{i-1} &i=j\\ h_j\partial_{i-1} & i>j+1; \end{cases}$$

$$ s_ih_j=\begin{cases}h_{j+1}\partial_i & i\leq j\\ h_js_{i-1} & i>j. \end{cases} $$

One way to try to do it is as follows.

Let $[\times 2]:\Delta\to\Delta$ denote the endomorphism "doubling" each linear order: $$[0\le 1\le \dotsb \le n] \longmapsto [0 \le 0' \le 1 \le 1'\le \dotsb \le n \le n'] $$ $$ f:[0\le 1\le \dotsb \le n]\to [0\le 1\le \dotsb \le m] \,\longmapsto\, f':[0 \le 0' \le 1 \le 1'\le \dotsb \le n \le n'] \to [0 \le 0' \le 1 \le 1'\le \dotsb \le m \le m']$$ $$ f'(i)=f(i), f'(i')=f(i)' \forall 0\leq i\leq n.$$ If I understand correctly the answer by Dmitri Pavlov, a standard argument gives that $Y_\bullet[\times 2]$ is a path object: $$Y_\bullet\xrightarrow{w_\bullet} Y_\bullet[\times 2]\xrightarrow{p_\bullet} Y_\bullet\times Y_\bullet$$ where each $w_n:Y_n \to Y_{2n+1}$ corresponds to the map of preorders $$[0 \le 0' \le 1 \le 1'\le \dotsb \le n \le n']\to [0 \le 1 \le \dotsb \le n] , i,i'\mapsto i $$ "repeating each coordinate twice", and the projections $p^\text{even}_\bullet,p^\text{odd}_\bullet:Y_\bullet[\times 2]\to Y_\bullet$ are maps "leaving only even/odd coordinates", i.e. are given by $p_n^\text{even}: Y_{2n+1}\to Y_n $ corresponding to $$[0 \le 1 \le \dotsb \le n]\to [0 \le 0' \le 1 \le 1'\le \dotsb \le n \le n'], i\mapsto i$$ and $p_n^\text{odd}: Y_{2n+1}\to Y_n $ corresponding to $$[0 \le 1 \le \dotsb \le n]\to[0 \le 0' \le 1 \le 1'\le \dotsb \le n \le n'], i\mapsto i'$$

**Question.** Is it true (under some “niceness” assumptions) that in a simplicial category
for $X_\bullet$ connected,
two maps $f_\bullet,g_\bullet:X_\bullet\to Y_\bullet$ are (strictly) homotopic
iff both maps $ X_\bullet\xrightarrow {f_\bullet} Y_\bullet$ and $ X_\bullet\xrightarrow {g_\bullet} Y_\bullet$
factor via $Y_\bullet[\times 2]$ as
$$ X_\bullet\xrightarrow {\tilde h_\bullet} Y_\bullet\circ[\times 2] \xrightarrow{p^\text{even}_\bullet} Y_\bullet$$
$$ X_\bullet\xrightarrow {\tilde h_\bullet} Y_\bullet\circ[\times 2] \xrightarrow{p^\text{odd}_\bullet} Y_\bullet$$
where $p^\text{even}_\bullet:X_\bullet\circ[\times 2]\to X_\bullet$ and $p^\text{odd}_\bullet:X_\bullet\circ[\times 2]\to X_\bullet$
are maps induced by “forgetting” the odd/even half of the “doubled” linear order $0\le 1 \le \dotsb \le n \le 0' \le 1'\le \dotsb\le n'$, as above.

This appears to be true if one of the maps is trivial (i.e. is a map to a point), see Defining homotopy via endofunctors of a simplicial category.