You might want to post your question again with the above target example added. There is a lot of work on combinig triples, e.g. Manes. My impression is that it is not obvious what it means to combine 2 triples. There is some mention of combining triples with distribute laws in Barr, Wells, toposes, triples, and theories, '85, Springer Bicategories may be relevant. There is a paper by Palmquist, in I think Springer LNM 195, on the double category of adjoint pairs.
Double categories are categories internal to the category of small categories.
2-categories are a special case of both bicategories and of double categories. Monoidal categories are a special case of bicategories. Borceux, handbook of categorical algebra, '94, Cambridge may be helpful on some of this. Finitary triples are those commuting with filtered colimits. The net result of a finite such stack can also be obtained with just 2 special layers.