Given a topological space $(X,\tau)$ we can define the "$T\_2$-ification" of $X$ by setting $T\_2(X,\tau) = X/\simeq$ where $x\simeq y$ in $X$ if and only if for every open neighborhood of $x$ has nonempty intersection with every open neighborhood of $y$. Then $T\_2(X,\tau)$ has the following universal property:


For every $T\_2$-space $Z$ and continuous function $f: X\to Z$, there is $\bar{f}: T\_2(X,\tau) \to Z$ such that $f = \bar{f} \circ pr$ where $pr: X\to X/\simeq$ is the canonical projection.


A *co-$T\_2$-ification* would be a space with a similar property as above, but all arrows reversed. (I hope this description is clear enough.)


Does every space have a co-$T\_2$-ification?


(There are similar constructions for $T\_i$-ifications at least for $i\in \{0,1\}$, so the same question could be asked for those.)



Co-Hausdorffification
The procedure you describe does not produce a $T\_2$-space in general. You can read [here](https://mathoverflow.net/questions/11191/nonhausdorff-dimension) and [here](https://mathoverflow.net/questions/78175/largest-hausdorff-quotient) on MO, and also [this nice recent bachelor thesis by Bart van Munster](http://www.math.leidenuniv.nl/scripties/BachVanMunster.pdf). But you do get the Hausdorffification by iterating that procedure (transfinitely many times, in general). It follows that $T\_2$-spaces form a reflective (full, replete) subcategory of $\mathbf{Top}$. There is no co-Hausdorffification though, because the only subcategory of $\mathbf{Top}$ that is simultaneously reflective and coreflective is $\mathbf{Top}$ itself: see Kannan, [Reflective cum coreflective subcategories in topology](http://dx.doi.org/10.1007/BF01419625), Math. Ann. 195 (1972), 168-174, [gdz](http://resolver.sub.uni-goettingen.de/purl?GDZPPN002305828), and also the 2006 PhD thesis by Martin Sleziak, available [here](http://thales.doa.fmph.uniba.sk/sleziak/papers/).


**Added:** Apologies if I misread the question by assuming the factorization to be unique. In order to partially rehabilitate myself let me add some comments. The full subcategory of $T\_i$-spaces happens to be reflective for $i=0,1,2,3$ (Kennison characterized reflective subcategories of $\mathbf{Top}$ as those full subcategories which are closed under products and subspaces in $\mathbf{Top}$, see his paper "[Reflective functors in general topology and elsewhere](https://doi.org/10.1090/S0002-9947-1965-0174611-9)", Trans. Amer. Math. Soc. (1965), 303-315). This means that for those values of $i$ there exists a notion of $T\_i$-ification (and the factorization is unique). Regarding the weak co-$T\_i$-ification (let us call the corresponding weak reflector $WR\_i$), at least for $i=1,2,3$ it does not exist by the argument in Eric Wofsey's beautiful answer. If you prefer a published account you can consider Lemma 5.5 in Herrlich's paper "[Almost reflective subcategories of $\mathbf{Top}$](http://dx.doi.org/10.1016/0166-8641(93)90115-T)", Topol. Appl. 49 (1993), 251-264, which states:


Lemma 5.5: If $\alpha$ is an infinite cardinal with successor $\alpha+1$, then there is a continuous map $f\colon 2^{\alpha+1}\to S$ from the $\alpha+1$ power of the discrete 2-point space $2$ onto the Sierpinski space $S$, which does not factor through any $T\_1$-space $X$ with $\mathrm{Card}(X)\leq\alpha$.


Then proceed as follows: if such a weak co-$T\_i$-ification $WR\_i$ existed, take $\alpha$ to be the cardinality of $WR\_i(S)$. Observe that $WR\_i(S)$ is $T\_1$ and $2^{\alpha+1}$ is $T\_i$, $i=1,2,3$, being a power of a $T\_i$-space, and apply the lemma.


I would be interested to know whether $T\_0$-spaces are weakly coreflective in $\mathbf{Top}$.


