Dual object

Remarks

Example: in $\mathbf{Vect}_k$, a finite-dimensional space $V$ with basis $e_1, \ldots, e_n$ has the dual space $V^*$ with dual basis $e^1, \ldots, e^n$ as dual. Take $\mathrm{ev}(f \otimes v) = f(v)$ and $\mathrm{coev}(1) = \sum_i e_i \otimes e^i$. The first identity sends $v$ to $\sum_i e_i \, e^i(v) = v$, and the second sends $f$ to $\sum_i f(e_i)\, e^i = f$. Non-example: an infinite-dimensional space $V$ has no dual in $\mathbf{Vect}_k$. The element $\mathrm{coev}(1)$ is a finite sum $\sum_{i=1}^m v_i \otimes w_i$, so the first identity would express every $v \in V$ as a combination of $v_1, \ldots, v_m$, which is impossible. Duals are unique up to unique isomorphism compatible with $\mathrm{ev}$ and $\mathrm{coev}$. In a symmetric monoidal category left and right duals coincide, and the trace of $f \colon A \to A$ is $\mathrm{ev} \circ \gamma_{A,A^*} \circ (f \otimes \id_{A^*}) \circ \mathrm{coev} \colon I \to I$.