Two maps $f,g$ into $Y$ are n-homotopic if, for every complex $K$ of dimension at most $n$ and for every map $\phi$ of $K$ into $X$ the compositions $f \phi,g\phi:K \to Y$ are homotopic.
As a sort of converse I am trying (it is an exercise in Mosher and Tangora, and hence I am confident they exist!) to find example of spaces $X$ and $Y$ and maps $f,g:X \to Y$ such that $f \phi \sim g \phi$ for any map $\phi$ of any complex $K$ into $X$ with with $f,g$ not homotopic.
Now I can never seem to develop counter-examples, and I guess I would like to understand the thought process that one goes through when developing these. So here my thought process is that $\phi$ can basically be any complex and any map, so I best concentrate my efforts on $X$ and $Y$. Initially I started thinking about the usual 'warning space' one sees with the Whitehead theorem (e.g $S^2 \times \mathbb{R}P^3$ and $\mathbb{R}P^2 \times S^3$). But this is specifically an example where we don't have a map $f:X \to Y$, so probably not relevant here. I thought some sort of maps like $S^1 \vee S^1 \to S^1$ where $f$ and $g$ collapse either one of the circles might work, but I don't think so either...
So - I'm not looking for an answer necessarily, but rather - when you see this question, what jumps into your head as a way to come up with a counter example?