  This paper concerns the final value problem for the heat equation under the
homogeneous Neumann condition on the boundary of a smooth open set in Euclidean
space. The problem is here shown to be isomorphically well posed in the sense
that there exists a linear homeomorphism between suitably chosen Hilbert spaces
containing the solutions and the data, respectively. This improves a recent
work of the author, in which the problem was proven well-posed in the original
sense of Hadamard under an additional assumption of H\"older continuity of the
source term. The point of departure is an abstract analysis in spaces of vector
distributions of final value problems generated by coercive Lax--Milgram
operators, yielding isomorphic well-posedness for such problems. Hereby the
data space is the graph normed domain of an unbounded operator that maps final
states to the corresponding initial states, resulting in a non-local
compatibility condition on the data. As a novelty, a stronger version of the
compatibility condition is introduced with the purpose of characterising the
data that yield solutions having the regularity property of being square
integrable in the generator's graph norm (instead of its form domain norm).
This result allows a direct application to the considered inverse Neumann heat
problem.
