Coends and the tensor product of $\mathcal{C}$-modules

We give an introduction to the concept of Kan extensions, and study its relation with the notions of coend and adjoint functors. We state and prove in detail a well known formula to compute Kan extensions by using coends: a certain colimit related to the concept of copower. Finally, we study the tensor product of functors, and its relation with Kan extensions, in order to represent the tensor product of $\mathcal{C}$-modules as a particular case.
Comment: 37 pages, 42 figures. Some terminology mistakes in the first version were fixed. Typos corrected. The formula to compute Kan extensions using coends was slightly generalized from small categories to skeletally small categories. Versions 2 and 1 in Spanish. Versions 2 and 3 are the same but in different languages