TITLE: How do I formulate this under ZFC set theory?
QUESTION [6 upvotes]: Here is a lemma given in Munkres-Elements of algebraic topology

This statement seems impossible to be encoded in ZFC set theory. The condition $(a)$ can be formulated in ZFC, but $(b)$ seems impossible to be formulated in ZFC. Could it be encoded in ZFC set theory?
For example, since there is a complete explicit way to construct the singular homology groups of topological spaces, saying "there is a homology functor $H_n:Top\rightarrow Ab$" can be completely formulated in ZFC. However, the condition $(b)$ asserts that there exists a class $\{D_X:X\in Top\}$ such that $D_X$ is natural for all $X$. This cannot be formulated in ZFC. I wonder if there is a smart trick to avoid this size problem.

REPLY [6 votes]: Yes, this can be expressed in ZFC. The issue, as you state, is that the "map" $D:X\mapsto D_X$ is a class function, since there are proper class-many topological spaces. However, this is fine: the map $D$ is definable by a (very messy) formula $\psi$ in the language of set theory, so, rather than writing e.g. "$D_X$ satisfies ...", we can instead write "For every $A$, if $\psi(A, X)$ holds, then $A$ satisfies . . .". (Why is $D$ definable? Look at the proof of the lemma!)
Of course, the general treatment of class maps does go beyond ZFC, either to a class theory like $NBG$ or $MK$, or to ZFC augmented with appropriately large sets, e.g. ZFC+Universes. But this is unnecessary here, and in most contexts.