mapReduce collection The name of the collection on which you want to perform map-reduce.
So — every document in this collection is given as an input to the "map" step, or the "query" step if there is one? Consider making this explicit.
You may output to a collection when performing map reduce operations on the primary members of the set; on secondary members you may only use the inline output.
By "set" do you mean "replica set"?
The formatting is a bit off. "If false" is outside of a blockquote, but "if true" is inside of the "if false" blockquote instead of being parallel to it. Also the final clause "The jsMode defaults to false." should be outside of the block quote.
The following is a prototype usage of the mapReduce command:
Is this intended to be a prototype usage, with blanks for people to fill in? Or an example of usage, something which could be typed in and run as-is? It's not actually either one. The mapFunction omits the body code, so it can't be an example. The query has code, so it can't be a prototype.
There's an actual example later, so perhaps this should be an actual prototype.
The map function exhibits the following behaviors:
The list following this is mostly requirements not behaviors, so it might be better to say "The map function has the following requirements"
The reduce function exhibits the following behaviors:
As with the map function, this is mostly requirements
the type of the return object must be identical to the type of the value emitted by the map function to ensure that the following operations is true:
Non-sequitur, since having identical types does not ensure that the following item is true. Both clauses are important — the only problem is the connection between them, which is given as "because" but needs to be "and".
Break this into two bullet points, one about the types and one showing the equation.
(Also, it's not a "following operations". It's a "following equation")
Also also, if you're using words like "idempotent", the first equation could be called "associative", and the third one "commutative".