Algorithms and procedures to recover objects and morphisms from the properties of category?

Question

Let C be a category and let us have many diagrams, that give properties of this category, e.g. existence of limits, existence of pentagon identities, existence of monoidal or other algebra like structures. So, are there algorithms or procedures, how can we recover the more or less concrete content of objects and morphisms of this category? Take for example Institution, which is structure, that characterizes syntax and all possible models of some logic. This structure involves the category of signatures. Can we use properties of institution to recover signatures. I.e. can we define category and only after then recover the objects and morphisms?

Answer 1

Based on my limited knowledge in the area, and judging by typical expert responses to the question so far, this issue seems to be completely unanticipated by the existing expertise, and by all likelihood, there's no body of knowledge dedicated specifically to the kind of problem that it presents.

You may want to have a look at Computational Category Theory by Rydeheard and Burstall, if you haven't already. For what I remember of it, its approach might be somewhat askew for your purpose, but a lot of it would probably apply.