A note on functor \((\cdot)^\circ\) (Q2774170)

Let \(A\) be an algebra over a field \(k\) and let \(A^*\) be the dual space. Then \(A^*\) contains a subspace \(A^\circ\) which is a coalgebra, called the finite dual of \(A\). If \(f\colon A\to B\) is an algebra map, then \(f^*\colon B^*\to A^*\) satisfies \(f^*(B^\circ)\subseteq A^\circ\). This paper first gives an example showing that \(f^*(B^\circ)\subseteq A^\circ\) is not true for any \(k\)-linear map \(f\colon A\to B\). It is well-known that the natural map \(\xi_A\colon A\to A^{\circ*}\) is an algebra map. If \(\xi_A\) is an algebra isomorphism then \(A\) is called reflexive. Let \(C\) be a coalgebra and let \(A\) be a reflexive algebra. It is shown that the natural \(k\)-linear isomorphism \(\Hom(C,A)\cong\Hom(C,A^{\circ*})=\Hom(C,\Hom(A^\circ,k))\cong\Hom(C\otimes A^\circ,k)=(C\otimes A^\circ)^*\) is an algebra isomorphism.