Let $f,g: X\_{\bullet}\to Y\_{\bullet}$ be two simplicial maps between simplicial sets. We say $f$ and $g$ are (strictly) simplicial homotopic if there exists a simplicial map 
$H: X\_{\bullet}\times I\_{\bullet}\to Y\_{\bullet}$ such that 
$$
f=H\circ\varepsilon\_0 \text{ and } g=H\circ\varepsilon\_1
$$
where $\varepsilon\_{\mu}: X\_{\bullet}\to X\_{\bullet}\times I\_{\bullet}$, $\mu=0,1$ are the two obvious inclusions.


We also know that there is a combinatorial definition of simplicial homotopy between simplicial maps: 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.


1. $$\partial\_0 h\_0=f\_p, \partial\_{p+1}h\_p=g\_p;$$
2. $$
\partial\_ih\_j=\begin{cases}h\_{j-1}\partial\_i & i<j\\
\partial\_ih\_{i-1} &i=j\\
h\_j\partial\_{i-1} & i>j+1
\end{cases};$$
3. $$
s\_ih\_j=\begin{cases}h\_{j+1}\partial\_i & i\leq j\\
h\_js\_{i-1} & i>j
\end{cases}.
$$



> 
> My question is: could we find in the literature that these two versions of simplicial homotopy are equivalent?
> 
> 
> 



Where to find the proof that these two version of simplicial homotopy are equivalent?
Proposition 6.2 in Chapter 1 of "Simplicial objects in algebraic topology", by J.P. May.


