Let $\mathcal{H}$ denote the homotopy category associated with the Kan-Quillen model structure on $\mathbf{sSet}$. Suppose we have a map $f\colon X \to Y$ between Kan complexes, such that for every finite simplicial set $K$ we have an isomorphism of the form: $$\mathcal{H}(K,X) \cong \mathcal{H}(K,Y)$$ induced by postcomposition with $[f]$. Is it true that $f$ is then a weak equivalence?



Do finite simplicial sets jointly detect isomorphisms in the homotopy category?
The answer is no. Otherwise, it would follow from Brown's representability theorem (and here I mean very specifically Theorem 2.8 from Brown's 1965 paper *Abstract Homotopy Theory*) that every "half-exact" functor on the homotopy category of unbased simplicial sets is representable. That is however false by my answer [here](https://mathoverflow.net/a/105049/12547). See also this [post](https://golem.ph.utexas.edu/category/2012/08/brown_representability.html) and the discussion below for some context.


