This case is easy and just like for 1-categories.
Define the support of an object to be the image of the unique morphism . That, is is an eso-ff factorization. Since is ff, is subterminal, and since esos are orthogonal to ffs, it is a reflection into .
Perhaps surprisingly, the next easiest case is the posetal reflection.
In any (1,2)-exact 2-category the inclusion of the posetal objects has a left adjoint called the (0,1)-truncation.
Given , define to be the (ff) image of . Since esos are stable under pullback, is a homwise-discrete category, and it clearly has a functor from , so it is a (1,2)-congruence. Let be its quotient. By the classification of congruences, is posetal. And if we have any where is posetal, then we have an induced functor . But is posetal, so is a (1,2)-congruence, and thus factors through a functor . This then equips with an action by the (1,2)-congruence , so that it descends to a map . It is easy to check that 2-cells also descend, so is a reflection of into .
This is actually a special case of the (eso+full,faithful) factorization system?, since an object is posetal iff is faithful. The proof is also an evident specialization of that.
The discrete reflection, on the other hand, requires some additional structure.
Given , define to be the equivalence relation generated by the image of ; this can be constructed with countable unions in the usual way. Then is a 1-congruence, and as in the posetal case we can show that its quotient is a discrete reflection of .
There are other sufficient conditions on for the discretization to exist; see for instance classifying cosieve. We can also derive it if we have groupoid reflections, since the discretization is the groupoid reflection of the posetal reflection.
The groupoid reflection is the hardest and also requires infinitary structure. Note that the 2-pretopos does not admit groupoid reflections (the groupoid reflection of the “walking parallel pair of arrows” is ).