  We examine the interplay between projectivity (in the sense that was
introduced by S.~Ghilardi) and uniform post-interpolant for the classical and
intuitionistic propositional logic. More precisely, we explore whether a
projective substitution of a formula is equivalent to its uniform
post-interpolant, assuming the substitution leaves the variables of the
interpolant unchanged. We show that in classical logic, this holds for all
formulas. Although such a nice property is missing in intuitionistic logic, we
provide Kripke semantical characterisation for propositions with this property.
  As a main application of this, we show that the unification type of some
extensions of intuitionistic logic are finitary. In the end, we study
admissibility for intuitionistic logic, relative to some sets of formulae.
  The first author of this paper recently considered a particular case of this
relativised admissibility and found it useful in characterising the provability
logic of Heyting Arithmetic.
