  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.
