Construction of coend and the reconstruction theorem of bialgebras

Abstract

Assume $k$ is a field and let $F:C\rightarrow Vect_{k}$ be a small $k$-linear functor from a $k$-linear abelian category $C$ to the category of vector spaces over the field $k$, the purpose of this note is to use a little knowledge of linear algebra and category to give the description of $end(F)$ and $coend(F)$, and then we give the reconstruction theorem of bialgebras by using this description. We use a constructive approach to understand $end(F),coend(F)$ and we describe the bialgebra structure of $coend(F)$ concretely when $F$ is a tensor functor.