Is there a name for isomorphism-preserving functions in the sense that whenever $x,y$ are sets of the same cardinality, so are $f(x)$ and $f(y)$?
For instance, working within the NBG set theory, denote by $\mathbf{S}$ the class of sets, and consider the function $f:\mathbf{S}\rightarrow\mathbf{S}$ satisfying for every $x \in \mathbf{S}$, $f(x) = \mathbb{P}x$. Then $f$ is isomorphism-preserving, since whenever $y, z \in \mathbf{S}$ have the same cardinality, so do their powersets.
On the other hand, let $a, a', b \in \mathbf{S}$ be such that $a$ and $a'$ are disjoint sets of the same, non-zero cardinality, and let $g:a\rightarrow b$. Consider the function $f':\mathbf{S}\rightarrow\mathbf{S}$ satisfying for every $x \in \mathbf{S}$, $f'(x) = g[x]$, i.e. $f'(x)$ is the image of $x$ under $g$, which, by the replacement axiom, is a set. Then $$ |f'(a)| = |g[a]| \neq 0 = |\emptyset| = |g[a']| = |f'(a')|. $$