Fix a logic $L$ and consider the category $\mathbf{AlgTh}_L$ with theories of $L$ as objects and theory interpretations as morphisms. For nice enough logics, this category has pushouts (which we will assume is the case).

Question: how large a 'graph' has been actually built using this (old) idea? There are plenty of flat lists of objects of such a category, for example Wikipedia has two: algebraic structure and list of algebraic structures (with lots of overlap as well as being non-trivially different), Jipsen has a list (not all of which are properly defined) and a graph with 48 entries. One can find other partial graphs too: page 136-141 of the CASL reference manual or the specialized-for-groups graph of Coq Theories required to prove the Feit-Thompson theorem. There are even similar lists for (modal) logics. But I have yet to find a decent (categorical) graphs. Has it not been done?

[I am interested in this because I am building such a graph, and I don't want to waste my time inventing things that have already been done. Also, I want to be as complete as I can.]