If $A,B$ are non-unital commutative algebras over a field $R$, what should their (categorical) coproduct?
I know for unital algebras the tensor product is the coproduct, but I think the construction of tensor product as a coproduct only worked there because we could have a unit element in multiplication, so we could just send, for example, $a\otimes 1_B \to f(a)g(1_B)=f(a)$ for some morphisms $f:A\to C$, $g:B\to C$ where $C$ is another algebra.
Apparently without unit element we can't do this anymore. So what would coproduct be in this case?
The coproduct of $A$ and $B$ has underlying $R$-vector space $A \oplus B \oplus (A\otimes B)$ and with multiplication defined by bilinearity and by $$(a_1,b_1,a'_1\otimes b'_1) \cdot (a_2,b_2,a'_2\otimes b'_2)= (a_1\cdot a_2,b_1\cdot b_2,a_1\otimes b_2 + a_2\otimes b_1 + (a_1\cdot a'_2)\otimes b'_2+(a'_1\cdot a_2)\otimes b'_1 + a'_2\otimes(b_1\cdot b'_2) + a'_1\otimes (b'_1\cdot b_2) + (a'_1\cdot a'_2)\otimes (b'_1\cdot b'_2))$$ for $a_1,a_2,a'_1,a'_2$ in $A$ and $b_1,b_2,b'_1,b'_2$ in $B$. The the first coproduct inclusion sends $a$ to $(a,0,0)$ and the second sends $b$ to $(0,b,0)$.