before i answer your question let me give the definition of product and coproduct for the most general case, i.e. for a "family"

of objects in a category

:

1) the product is an objcet

and morphims

such that for every object

and morphisms

there exists a unique morphism

such that

2) the coproduct is an object

and morphisms

such that for every object

and morphisms

there exists a unique morphism

such that

__from now on__ i'll assume that

is the category of all vector spaces over a field

[or the category of abelian groups or in general the category of

modules, where

is a

(commutative) ring]. then, exactly as i showed in my previous post,

will be the product of

clearly if

is the category of

__finite dimensional__ vector spaces over

a field

then the index set

has to be finite because otherwise the vector space

will not be in

anymore.

__for the coproduct__ let

and define

to be the natural injection. let

be the natural projection. now for any

and

morphisms

define

by

this definition is well-defined only if only we have finitely many term in the sum, which explains why we've

considered

instead of

(this also answers

**aliceinwonderland**'s question!) now for any

we have

thus

for all

so, in order to prove that

is the coproduct of

we only need to prove the uniqueness of

which is easy.

again, note that if

is the category of

__finite dimensional__ vector spaces over some field

then the index set

will have to be finite.