It is well-known that partial orders can be seen as very simple categories (those where there is at most one morphism between every two objects).
Then, the notion of "(binary) join of two elements (i.e, the minimum of the upper bounds of these two elements)" can be understood in categorical terms; indeed it corresponds to the coproduct (and so it is a particular case of the colimit construction).
Is there some (well-known?) categorical notion which captures, when constrained to partial orders, the notion of being a minimal element of the upper bounds of these two elements?
I can see some reasonable ways to generalize the definition of the minimality to an (abstract) categorical setting, but either I am considering a very weak notion (i.e., with a lot of instances) or coproducts do not satisfy my property. In both cases I do not feel that the proposals I have thought are reasonable generalizations to the categorical setting.
If I understand correctly you are looking for a minimal upper bound (whatever that means) instead of a minimum upper bound in a general category. So you can have 2 nonisomorphic minimal objects. This means your contruction cannot be a limit (at least not in an obvious way). Also :how do you define upper bound in a general category? One way to do this is to transform your general category into a preorder (all arrows from a to b go to a single arrow) and then take your minimal upper bounds.
So express categorically , if you can, "minimal upper bound in a preorder" and then precompose it with the thinning functor defined above Hth