Given a topological space $X$, we consider the following category $\mathsf{CW}_{X\to}$. The objects are finite CW complexes $Y$ equipped with a continuous map $X\to Y$. The morphisms are continuous maps $Y\to Y'$ such that the composition $X\to Y\to Y'$ agrees with $X\to Y'$. Now there is a functor: $$\mathsf{CW}_{X\to}\to\mathsf{AbGrp}$$ given by taking the cohomology of $Y$. I wish to consider the colimit of this functor: $$\operatorname{colim}H^\ast(\mathsf{CW}_{X\to})$$

Now let's assume that $X$ is compact Hausdorff. I believe that in this case $\operatorname{colim}H^\ast(\mathsf{CW}_{X\to})$ is naturally isomorphic to the Cech cohomology of $X$. However, I am a bit unsure as to whether I should take the category $\mathsf{CW}_{X\to}$ as I have defined it above, or "soften" it by taking only *homotopy classes* of maps $X\to Y$ and *homotopy classes* of maps $Y\to Y'$ (maybe we could call this category $\mathsf{CW}_{X\to}^h$).

How should I think about the relationship betwee $\operatorname{colim}H^\ast(\mathsf{CW}_{X\to})$ and $\operatorname{colim}H^\ast(\mathsf{CW}_{X\to}^h)$?

Is one easier to deal with theoretically (e.g. by virtue of being filtered)? Are they indeed isomorphic via the natural map: $$\operatorname{colim}H^\ast(\mathsf{CW}_{X\to}^h)\to\operatorname{colim}H^\ast(\mathsf{CW}_{X\to})$$ induced by the forgetful functor $\mathsf{CW}_{X\to}\to\mathsf{CW}_{X\to}^h$?