I was scrolling nlab instead of studying my topology final, and stumbled on the following page: https://ncatlab.org/nlab/show/separation+axioms+in+terms+of+lifting+properties#hausdorff_spaces_
And intrigued, I tried to understand the lifting property for Hausdorff spaces. Which I understand as follows:
Denote by $*$ the singleton topological space, by 2 the indiscrete space on two points $\{0,1\}$ and by A the space whose set is $\{0,1,2\}$ and whose open sets are $\{\emptyset, \{0\}, \{1\}, \{0,1\}, \{0,1,2\}\}$. We have the following categorical definition of X being Hausdorff: X is hausdorff is and only if for all $f:2\to X$, which makes the square commute, there exists a diagonal morphism, such that the diagram commutes 
I am almost sure I misunderstood this, because I am assuming all arrows should be continuous, but taking the indiscrete on 2, means the function from 2 to A just isn't. Should I just take the discrete instead of the indiscrete on 2? but that isn't what the nlab is leading me to believe. Am I misunderstanding the category in which this diagram lies?