As far as I understand, I cannot define the values of built-in functions: Sin[Cat] := Dog will result in error. This is because all built-in symbols have attribute Protected.
However for the function N[], which is also protected, I can define a custom value:
N[Cat] := Dog
By the way, doing so will not change DownValues[N].
So the question is: why function N[] behaves like that? Is it just some hard-coded rule, which is not described in terms of attributes, downvalues etc.?
Values of this type are actually stored in an analog to DownValues called NValues. Here's an example:
In[13]:= N[IndianaPi] = 4; IndianaPi // NValues Out[14]= {HoldPattern[ N[IndianaPi, {MachinePrecision, MachinePrecision}]] :> 4} As xzczd points out this is mostly a subset of what's mentioned here
Note that a similar mechanism is used for Format:
In[43]:= Format[IndianaPi] = Interpretation[I\[Pi], IndianaPi]; IndianaPi // FormatValues Per bcp's comment, the Protected attribute need not be a problem because no assignment to N is actually ever done. For instance:
In[118]:= A /: HoldPattern@Set[A[q_], v_] := (q -> v); In[119]:= Protect@A Out[119]= {"A"} In[120]:= A[1] = 2 Out[120]= 1 -> 2 That gives us no error. And yet:
In[121]:= Attributes@A Out[121]= {Protected} 
N's Protected attribute does not forbid me to write N[***] = ***? $\endgroup$ UpValues. I don't know how it's implemented internally. $\endgroup$
NValues. $\endgroup$