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 similary 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.)

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.)