Consider $\pi_*:Ho(Top_*)\to Set^{\mathbb N}$ defined by $[X,x]\mapsto \{\pi_n(X,x)\}_n$ from the homotopy category of the pointed topological spaces. I showed that this is a conservative functor and I want to show that it preserves limits.
So let $\{f_j:[X,x]\to [Y_j,y_j]\}$ be a limit in $Ho(Top_*)$, we have that $f_j$ can be written as the homotopy class of some $f_j^{cf}:(X,x)^{cf}\to (Y_j,y_j)^{cf}$ where we replace $X,Y_j$ by bifibrant replacements.
Now we get that $\{\pi_*(f_j):\{\pi_n(X,x)\}\to\{\pi_n(Y_j,y_j)\}\}$ satisfy the cone condition, the problem is to show the universal property.
So suppose $\{q_j:\{S_n\}\to \{\pi_n(Y_j,y_j)\}\}$ satisfies the cone condition, I guess the idea is to use the fact that $[X,x]$ is a limit but I don't know how to do it. Is it a left/right adjoint ?
Any ideas ?