Mariano already answered the question, but let me make two additional remarks:

1) Actually the proof of GAFT is *constructive* and can be (at least sometimes) used to get explicitly a left adjoint functor. In this case, you can see directly the "big coequalizer" whose set-theoretic existence issue is dealt with the solution set condition: just consider all maps from your space $X$ into Hausdorff spaces which are *surjective*; this bounds the cardinality of the Hausdorff spaces, and thus up to isomorphism there is only a set of them.

2) Another construction of the left adjoint to $\mathsf{Haus} \to \mathsf{Top}$ works as follows: Let $X$ be a topological space, and consider the equivalence relation $\sim$ generated by: If $x,y$ cannot be separated by disjoint open sets, then $x \sim y$. Then $H(X):=X / \sim$ has the property that every map from $X$ into a Hausdorff space uniquely factors through $X \to H(X)$. If $H(X)$ was Hausdorff, we would be done. But this is not always the case. Instead, we have to repeat this construction: $X \to H(X) \to H(H(X)) \to H(H(H(X))) \to \dotsc$, then take the colimit $H^{\omega}(X)$, and make again $H^{\omega}(X) \to H(H^{\omega}(X)) \to \dotsc ...$. You can continue this for every ordinal number. Since $X$ is a set and all these maps are quotient maps, at some stage we get an isomorphism, which is the desired Hausdorff quotient.

It is interesting *when* we arrive at this stage, see my question about the nonhausdorff dimension.