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.

- $$\partial_0 h_0=f_p, \partial_{p+1}h_p=g_p;$$
$$ \partial_ih_j=\begin{cases}h_{j-1}\partial_i & i<j\\ \partial_ih_{i-1} &i=j\\ h_j\partial_{i-1} & i>j+1 \end{cases};$$

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

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