Suppse $Y$ is a connected CW complex and $Y_n$ denotes the $n$-skeleton of $Y$. Suppose $X$ is a connected CW complex of dimension $n$. The inclusion $\iota \colon Y_n \to Y$ yields a map $[X,Y_n] \to [X,Y]$. Is this map surjective?
I was thinking as follows. For a map $X\to Y$ let be $X \to Y_n$ a cellular map homotopic to the previous one. Is then this map $X \to Y_n$ mapped onto the map $X\to Y$ by the inclusion $\iota$?