Also, I am pretty sure that you can get the co(limit) formulas which apply to posets of open sets of a space as a special case. If your categories are...

Note that those formula are for arbitrary enrichment, but because we're talking about Set enrichment, tensor is cartesian product and the hom-object i...

@matteocapucci I believe that your account of the adjoint triple on the presheaf categories is correct. To get an explicit (co)limit formula for them ...