Priority of symbols in a notation for projections We define an initial function called projection as $$I^k_i(n_1,\ldots,n_k) = n_i, \quad i \leq 1 \leq k, \quad k \in \mathbb{N}.$$

Suppose now that we want to define it in some programming language where we have no access to superscripting nor subscripting. Should we call it Iki or Iik?

My stand point is that it should be Iki so that we first describe the nature of function, that it comes from a set of functions that take k arguments. But, the other approach is also understandable because it emphasizes the purpose of function, that it returns the i-th coordinate.
It may also be that priorities don't go from left to right as I imagine it.

Is there any convention on priority of symbols in this type of notation?
No, there is no convention.

In some languages, there is an idiomatic name for the function that extracts the first element of the tuple (e.g., fst, hd) and/or the function that extracts the second element (e.g., snd).