Absolute beginner in simplicial homotopy theory (and category theory) here. Trying to understand the idea of skeletons and coskeletons of a simplicial set.
Let $\mathbf\Delta$ be the simplex category, $\mathbf S$ be the category of simplicial sets, and $X: \mathbf\Delta^{op} \to Set$ be a simplicial set. The n-skeleton of $X$ is defined to be the simplicial subset $sk_nX=\bigcup_{\tau \in Mor_{\mathbf S}(\Delta^n,X)}\tau(\Delta^n)$, where $\Delta^n$ is the standard n-simplex $Mor_{\mathbf\Delta}(-,[n])$.
I.e. we have $sk_nX([m])=\bigcup\tau(Mor([m],[n]))\subseteq X_m$ in particular we have $sk_nX([n])=X_n$ by Yoneda Lemma.
If $Y$ is another simplicial set, apparently there is the following bijection:
(1) $Mor(sk_nX,Y) \cong Mor(X_{\leq n},Y_{\leq n})$ where $X_{\leq n}$ is the restriction of $X$ to the full subcategory $\mathbf\Delta_{\leq n}^{op}$.
How is this constructed? If we have a natural transformation $\eta \in Mor_{\mathbf S}(sk_nX,Y)$, then $\eta_{n}$ certainly is a morphism (of sets) from $sk_nX([n])=X_n \to Y_n$. What about for $m \lt n$ and $m \gt n$? We have $\eta_m: sk_nX([m]) \subseteq X_m \to Y_m$. How do we extend the morphism to all of $X_m$? Conversely, if we have a natural transformation $X_{\leq n}\to Y_{\leq n}$, then for each $m \leq n$ we can restrict to a morphism $sk_nX([m]) \to Y_m$. What about for $m \gt n$?
The n-coskeleton of $X$ is defined to be the simplicial set $csk_n X$ such that $csk_nX([m])=Mor(sk_n \Delta^m,X)$. Again, there is a bijection:
(2) $Mor(X_{\leq n},Y_{\leq n}) \cong Mor(X,csk_nY)$. How is this constructed?