gn.general topology - Notion of finite dimensional simplicial space

OK I checked, how the adjoint functors looks like. Given any $Delta|_N $ simplicial space $X$. To define $L(X)$, we have to extend $X$ to the whole category $Delta$. I am just telling, what $L(X)$ does on $[N+1]$. Then you keep extending the functor in the same way:

$L(X)([N+1]):=(0,ldots,N)times X([N])/sim$, where the equivalence relation is given by
$(j,s_k(x))sim (k+1,s_j(x))$ for $0le jle kle N,xin X[N-1]$. The $i$-th degeneracy map is induced by the inclusion of the i-th summand. Using the relations in $Delta$ one can also define the face maps.

The right adjoint functor is given by
$M(X)([N+1]):= ( (x_0,ldots,x_{N+1})|partial_ix_j=partial_{j-1}x_imbox{ for } 0 le i < j le N+1 )subset prod_{i=0}^{N+1}X[N]$. The face maps are just the projections and one can define the degeneracy maps using the relations in $Delta$.

So let $X$ be a $Delta$-space. The natural transformation is given by
$L(R(X))([N+1])rightarrow M(R(X))([N+1])qquad (i,x)mapsto (partial_0 s_i(x),ldots,partial_{N+1} s_i(x))$.

Using the relations in $Delta$ one can show, that this map is injective. So the remaining question is, whether this map is an open map (considered as a map onto the image).