Sentence:  A <WSD>cone</WSD> is an object (the apex) and a natural transformation from a constant functor (whose image is the apex of the <WSD>cone</WSD> and its identity morphism) to a diagram functor. Its components are projections from the apex to the objects of the diagram and it has a “naturality triangle” for each morphism in the diagram. (A “naturality triangle” is just a naturality square which is degenerate at its apex side.) 
Meaning of the word marked with <WSD> tags:  An object "V" together with an arrow going from "V" to each object of a diagram such that for any arrow "A" in the diagram, the pair of arrows from "V" which subtend "A" also commute with it. (Then "V" can be said to be the cone's "vertex" and the diagram which the cone subtends can be said to be its "base".)