Some branches of mathematics admit calculi with whom one can do syntactical (language-like, grammatical) or geometric operations to arrive at certaing conclusions. The syntactical part (proof theory) of mathematical logic is the prominent example of such calculi, but the usual analysis have similary calculi as well (for derivates and integrals) and the variational calculi is still another example of syntactical rules for inferring mathematical conclusions. Effectively - whole computer algebra is world of calculi.
I have heard about string diagrams for the monoidal categories as the calculi for the category theory, https://link.springer.com/article/10.1007/s10485-018-9549-8 is generalization of such work. But are there any other calculi for the category theory?
Specifically - I would like to arrive at the framework that could allow one to deduce the functor that is optimal in some kind of sense (e.g. as minimal action principles guide the search for optimal functions) and that can be applied in https://en.wikipedia.org/wiki/Synthetic_differential_geometry and https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis. But I hope that this applied part of my question should not disturb anyone to give the general answer to my question - what calculi are there for the category theory?
