  This paper is a sequel to "Logical systems I: Lambda calculi through
discreteness". It provides a general 2-categorical setting for extensional
calculi and shows how intensional and extensional calculi can be related in
logical systems. We define Yoneda triangles as relativisations of internal
adjunctions, and use them to characterise universes that admit a notion of
convolution. We show that such universes induce semantics for lambda calculi.
We prove that a construction analogical to enriched Day convolution works for
categories internal to a locally cartesian closed category with finite
colimits.
