When J is the category with just two objects and only identity arrows (we call these categories discrete), then a colimit of a J-shaped diagram is called the coproduct of the two objects in the C. The limit is called their product. This language carries over to larger discrete diagrams.
When J is just a pair of parallel arrows between two objects, a limit over this diagram is called the equalizer of the arrows, and the colimit is called their coequalizer.
J is called a span if it consists of objects a, b, and c, and only two nontrivial arrows f:a->b and g:a->c. The limit of a span-shaped diagram is called the pullback of the two arrows mapped two, and also called the fiber product of the image of the two outer objects in homotopical contexts. The colimit of a J*-shaped diagram (called a cospan diagram) is called the pushout of the two arrows.
A limit of the empty diagram is called an initial object in the category
Because adjoints are unique up to isomorphism, so also are limits and colimits, being determined by adjoint functors.
(Exercise: show that a limit lim(F) of a diagram F:J->C determines a family of morphisms p_j: lim(F)->F(j) for each object j in J, so that, for every morphism f:j->k in J, F(f)∘(p_j)=p_k. A dual argument will show that there are similar morphisms i_j: F(j)->colim(F) so that for all f:j->k in J, (i_k)∘F(f)=i_j.)
In the last exercise we found that there are projections (p_j) out of the limit of a diagram into each object so that the entire diagram commutes. Furthermore, by the uniqueness of adjoints, any other object admitting such projections must admit a morphism into the limit that makes all of the proejctions commute with one another, and this morphism is unique with this property. Dually, a colimit admits a morphism into any other object with the proper injections (i_j), so that everything commutes. These properties are called universal.
A limit in a category is the colimit of the dual diagram in the dual category.