I will denote $(\lambda x\in D: f(x)) = \{(x;f(x))\,|\,x\in D\}$ for every set $D$ and a form $f$ dependent on the variable $x$.
For a function $z$ and a function $a$ we define $\operatorname{Move} ( z ; a ) = \lambda x \in \left\{ c \in U^{\operatorname{im} a} \, | \, c \circ a \in \operatorname{dom} z \right\} : z \left( x \circ a \right)$ where $U$ is a big enough set (e.g. a Grothendieck universe).
See https://math.stackexchange.com/questions/131737/modifying-every-argument-of-a-multivariable-function for a background of this formula.
I want to find a nice proof of the following statement:
Conjecture Let $\gamma$ is a set of functions with disjoint domains. Then $ \operatorname{Move} \left( \bigcup \gamma ; a \right) = \bigcup_{g \in \gamma} \operatorname{Move} \left( g ; a \right) . $