Sentence:  An <WSD>inverse limit</WSD> has “natural projections” which are restrictions of the projections of the Cartesian product (to a domain which is the <WSD>inverse limit</WSD>). The reason why the projections are described as “natural” would be the following: besides the functor from an index poset to the inverse system, there is another functor from the same index poset to the <WSD>inverse limit</WSD> of that system, this functor being a constant functor. Then there is a natural transformation from the constant functor to the <WSD>inverse limit</WSD>'s functor: the components of such natural transformation are the said “natural projections”. 
Meaning of the word marked with <WSD> tags:  A subset of the Cartesian product of all the members of an inverse system, such that a member "M" of the subset is a sort of “cross section” of the inverse system (as fiber bundle) induced by the morphisms of it. (If i \le j in the indexing poset then f_{ij}:A_j \rightarrow A_i in the inverse system and if a_i \in A_i, a_j \in A_j are components of "M" then f_{ij}(a_j) = a_i).