Given a category $\mathbf{C}$, we can consider monomorphisms into it. These are the faithful and injective-on-objects functors (this violates the principle of equivalence). The idea is to try to get a lattice of subobjects.

Pullbacks (in the 1-categorical sense, which also has problems with principle of equivalence) preserve monomorphisms, so we can have a notion of "meet". Then, given monomorphisms $\mathbf{X} \to \mathbf{C}$ and $\mathbf{Y} \to \mathbf{C}$, we can take the image of the coproduct $\mathbf{X} + \mathbf{Y} \to \mathbf{C}$ and have a notion of "join". Here the image is the smallest subcategory containing all the images of morphisms under the functor (again, not really treating $\mathbf{Cat}$ as a 2-category).

Is this construction plausible or is there any problem I am overlooking? Does that poset have any extra structure? I am not sure if I should understand from here that for it to be a distributive lattice we would need $\mathbf{Cat}$ to be coherent, but it is not even regular. If not, can we impose some conditions on $\mathbf{C}$ to get some structure?