Let $C$ be the category of unital, commutative, $k$ algebras, where $k$ is a field.
Given a discrete category $I$, with two elements, let $F: I \to C$ be a functor.
Denote $I(i_1) = A_i$ for $i = 1,2$.
What's the colimit of this diagram?
My attempt:
The colimit, $colim_I F$, is $A_1 \otimes A_2$.
It can be shown that the two projections $p_i: A_i \to A_1 \otimes A_2$ given by $p_1(a_1) = a_1 \otimes 1, p_2(a_2) = 1 \otimes a_1$ are ring homomorphisms using the fact that $(a_1 + b_1, 1)$ ~ $ (a_1 , 1) + (b_1, 1)$.
So now I need to prove that given $W \in C$ and maps $f_i: A_i \to W$ in $C$, there is a unique map $f: A_1 \otimes A_2 \to W$ s.t $f \circ p_i = f_i$.
The uniqueness part seems straightforward to me. It is the issue of existence that I'm having trouble with:
I want to define $f$ using elements of the basis to $A_1 \otimes A_2$. Let's say that $\{e_j: j \in J\}$ and $\{f_k: k \in K\}$ are bases for $A_1, A_2$ respectively. Define $f(e \otimes f) = f_1(e)f_2(f)$ so that $f \circ p_i = f_i$ holds. It is straighforward that $f$ preserves the multiplicative structure on $A_1 \otimes A_2$ and the multiplicative identity. How can I show it preserves the additive structure? Is this the correct definition for $f$?