Tensor algebra $T(A \oplus B)$ is a coproduct of $T(A)$ and $T(B)$ in the category of unitary associative $(R, R)$-algebras.

Question

Let $A$ and $B$ be unitary $(R, R)$-bimodules. Let $f_1 \colon T(A) \to C$ and $f_2\colon T(B) \to C$ be identity preserving $(R, R$)-algebra homomorphisms. I need to define morphisms $\iota_1 \colon T(A) \to T(A \oplus B)$ and $\iota_2 \colon T(B) \to T(A \oplus B)$. My intuition told me that they are the inclusion maps. Not sure if this is right. How do I find a unique morphism $\bar{f} \colon T(A \oplus B) \to C$ such that $\bar{f} \circ \iota_i = f_i$.

Answer 1

Any left adjoint preserves colimits (MacLane, Categories for the working Mathematician, Theorem V.5.1). Now use that $T(-)$ is left adjoint to the forgetful functor from the category $(R,R)$-algebras to the category of $(R,R)$-modules. In fact, this holds for any monoidal category with $\otimes$-compatible coproducts (loc. cit., Theorem VII.3.2), where we apply this here to the monoidal category of $(R,R)$-modules aka $R$-bimodules.