If I define a diffeomorphism $\phi: \Sigma_{a} \rightarrow \Sigma_{b}$, can I also write $\phi: \Sigma_{a} \backslash A \rightarrow \Sigma_{b} \backslash B$ (for some appropriate $A$, $B$) and mean correctly the same $\phi$ operating on a restricted domain, and how, given some definition of $\phi$ do I state that $\Sigma_{d}$ is not diffeomorphically related to $\Sigma_{c}$ by $\phi$? Do I have to say e.g. $\phi:\Sigma_{c} \rightarrow \Sigma_{c}' \neq \Sigma_{d}$
Or, to ask the question another way, how do I define a diffeomorphism to mean a function that can be applied to different domains and state whether, given two particular $\Sigma$, $\phi$ is or is not a diffeomorphism between them?
What's the right terminology and notation here?
This question provides a good answer in the context of homeomorphisms, and since diffeomorphisms are suitably differentiable homeomorphisms the same should apply.
In brief, for some diffeomorphism $\phi:A \rightarrow B$, and subsets $A' \subset A$ and $B'=\phi(A') \subset B$, one defines the restriction of $\phi$ as e.g. $\phi':A \rightarrow B$ given by $\phi'(a) = \phi(a) \ \forall a \in A'$
One may also write a restriction as e.g. $\phi |_{A'}$
Given such a definition in terms of the image of a subset for which the original diffeomorphism is $\phi$ defined, the restriction $\phi'$ is also necessarily a diffeomorphism, or, to quote from the chapter "Smooth Maps" here (from J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, Springer)
"The restriction of a diffeomorphism to an open submanifold with or without boundary is a diffeomorphism onto its image."