Let $\mathbf S$ be the category of simplicial sets, $X$ and $Y$ simplicial sets, $X_{\leq n}$ the restriction of $X$ to the full subcategory $\mathbf \Delta_{\leq n}$ of the simplex category.
There is a map
$Mor_{\mathbf S}(X,Y) \to Mor_{\mathbf S}(X_{\leq n},Y_{\leq n})$
given by the restriction of natural transformations $X \to Y$ to the simplices of $X$ of degree $\leq n$.
Under what conditions do we get an inverse to this map (when any map $X \to Y$ is fully specified by its values on the lower degree simplices)?