Definition: A declared functional object

A declared functional object consists of

where $\mathcal{D}$ is its domain, $\mathcal{V}$ its target, $F:\mathcal{D}\to \mathcal{V}$ its functional, $\mathcal{A}$ its auxiliary data, and $\mathcal{E}$ its defining equations and boundary or update conditions.

A literal functional identification requires the same declared domain and target, with equality of the functionals. An identification between different domains additionally requires specified comparison maps and an equality after those maps. Equality of one evaluation is not either identification.