Some contributions to presheaf model theory, II -- back and forth

We discuss the back and forth technique in the context of presheaf model theory. The essence of the back and forth technique lies in showing the relationship between various hierarchies which calibrate similarity between two models and, more generally, between two pairs consisting of a model and a tuple from it. In this paper we define several such hierarchies for presheaf models (and tuples of sections from them): those based on the degree of extendibility of partial isomorphisms through literal back and forth conditions, on sharing specific, abstract invariants which we define (the $F^\alpha_{M,\bar{a}}$ of \S{}\ref{function_analysis} for example), on agreeing on the (truth) values of instantiations of formulae up to a given amount of quantifier completity, on the existence of winning strategies for player II in certain Ehrenfeucht-Fra\"iss\'e-type games and, finally, on satisfying certain infinitary sentences that arise in the construction of Scott sentences. We ultimately show that all of these hierarchies align.