= Solution
The syntax operation $(s,t)\mapsto\#(ST)$, where $s=\#S,t=\#T$, is a <total computable function> on valid codes: decode the finite syntax trees, form their application, and encode it again. Extend it arbitrarily on invalid codes if necessary. Similarly $n\mapsto\#(c_n)$ is total computable because the finite numeral syntax can be generated effectively from $n$. The preceding characterization of <lambda-definable functions> supplies closed representatives $A,C$ with
$$
\boxed{A d(S)d(T)\equiv_\beta d(ST),\qquad Cc_n\equiv_\beta d(c_n).}
$$
Here $d(S)=c_{\#S}$ is <lambda quotation>, not a code of a <beta equivalence> class. In particular $C$ quotes the syntax of the numeral itself; it is not simply successor or an identity operation.
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