  We prove first-order naturality of involutive Heegaard Floer homology, and
furthermore construct well-defined maps on involutive Heegaard Floer homology
associated to cobordisms between three-manifolds. We also prove analogous
naturality and functoriality results for involutive Floer theory for knots and
links. The proof relies on the doubling model for the involution, as well as
several variations.
