Dual basis, an Categorical approach

We already talk about the natural isomorphism(Riese Representation Theorem) between dual space functor and adjoint map. Math Essays: A Categorical Perspective on the Riesz Representation Theorem and Adjoint Maps (marco-yuze-zheng.blogspot.com)

I would like to talk about a categorical understanding of dual basis in this blog.

Let $M$$M$ be a finite rank free module over a commutative ring $R$$R$, the dual module functor is given by ${\mathrm{Hom}}_{R-\mathrm{Mod}}(-,R)$$\mathrm{Hom}_{R-\mathrm{Mod}}(-,R)$.

Let $e_1,...,e_n$$e_1,...,e_n$ be a basis of $M$$M$, then $M\cong R{e}_1\oplus ...\oplus R{e}_n$$M\cong Re_1\oplus...\oplus Re_n$. Notcie that $\mathrm{Hom}$$\mathrm{Hom}$ functor will map colimit to limit, hence

$$\begin{array}{}\text{(1)}& {\mathrm{Hom}}_{R-\mathrm{Mod}}(M,R)\cong {\mathrm{Hom}}_{R-\mathrm{Mod}}(R{e}_1\oplus ...\oplus R{e}_n,R)\cong \prod _{i=1}^n\mathrm{Hom}(R{e}_i,R)\cong \underset{i=1}{\overset{n}{⨁}}{\mathrm{Hom}}_{R-\mathrm{Mod}}(R{e}_i,R)\end{array}$$

Now, it is natural to pick ${\psi }^i\in {\mathrm{Hom}}_{R-\mathrm{Mod}}(R{e}_i,R)$$\psi^i\in\mathrm{Hom}_{R-\mathrm{Mod}}(Re_i,R)$ satisfy

$$\begin{array}{}\text{(2)}& {\delta }_{ij}={\psi }^i(e_j)\end{array}$$

to be the dual basis.