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

Construction of coend and the reconstruction theorem of bialgebras