Justification logics are an explication of modal logic; boxes are replaced with proof terms formally through realisation theorems. This can be achieved syntactically using a cut-free proof system e.g. using sequent, hypersequent or nested sequent calculi. In constructive modal logic, boxes and diamonds are decoupled and not De Morgan dual. Kuznets, Marin and Straßburger provide a justification counterpart to constructive modal logic CK and some extensions by making diamonds explicit by introducing new terms called satisfiers. We continue the line of work to provide a justification counterpart to Fischer Servi's intuitionistic modal logic IK and its extensions with the t and 4 axioms. We: extend the syntax of proof terms to accommodate the additional axioms of intuitionistic modal logic; provide an axiomatisation of these justification logics; provide a syntactic realisation procedure using a cut-free nested sequent system for intuitionistic modal logic introduced by Straßburger.
Marin et al. (Sat,) studied this question.