Can the lambda functions in Church numbers be swapped? I've learned that one can represent natural numbers with lambda calculus like this:

\begin{align*}
c_0 &amp;= \lambda s. \lambda z. z\\
c_1 &amp;= \lambda s. \lambda z. s~z\\
c_2 &amp;= \lambda s. \lambda z. s~(s~z)\\
c_3 &amp;= \lambda s. \lambda z. s~(s~(s~z))\\
\end{align*}

But could one also write

\begin{align*}
c'_0 &amp;= \lambda z. \lambda s. z\\
c'_1 &amp;= \lambda z. \lambda s. s~z\\
c'_2 &amp;= \lambda z. \lambda s. s~(s~z)\\
c'_3 &amp;= \lambda z. \lambda s. s~(s~(s~z))\\
\end{align*}

?

Why / why not?
Sure, except it's going to be more painful to use the numbers with the swapped arguments because in many cases you want to apply a Church numeral only to the argument $s$ and leave $z$ specified. Thus, instead of writing $c_n \, e$ you will have to write $\lambda z . c'_n \, z \, e$ whenever that happens.