  Many recent writers in the philosophy of mathematics have put great weight on
the relative categoricity of the traditional axiomatizations of our
foundational theories of arithmetic and set theory (\cite{Parsons1990a},
\cite{Parsons2008} \S{49}, \cite{McGee1997aa}, \cite{Lavine1999aa},
\cite{Vaananen2014aa}). Another great enterprise in contemporary philosophy of
mathematics has been Wright's and Hale's project of founding mathematics on
abstraction principles (\cite{Hale2001}, \cite{Cook2007aa}). In
\cite{Walsh2012aa}, it was noted that one traditional abstraction principle,
namely Hume's Principle, had a certain relative categoricity property, which
here we term \emph{natural relative categoricity}. In this paper, we show that
most other abstraction principles are \emph{not} naturally relatively
categorical, so that there is in fact a large amount of incompatibility between
these two recent trends in contemporary philosophy of mathematics. To better
understand the precise demands of relative categoricity in the context of
abstraction principles, we compare and contrast these constraints to (i)
stability-like acceptability criteria on abstraction principles (cf.
\cite{Cook2012aa}), (ii) the Tarski-Sher logicality requirements on abstraction
principles studied by Antonelli \cite{Antonelli2010aa} and
Fine~\cite{Fine2002}, and (iii) supervaluational ideas coming out of Hodes'
work \cite{Hodes1984, Hodes1990aa, Hodes1991}.
