Isomorphism of $A^{op} \otimes A$ and $End_k(A).$

Question

My professor is trying to show that the map $\phi: A^{op} \otimes A \rightarrow End_k(A)$ is an isomorphism for $A$ a CSA over $k.$ The map is $a \otimes b \mapsto (c \mapsto bca).$ However, is $(c \mapsto bca)$ an endomorphism of $A?$ I see how the map respects scalar multiplication of $k$ and addition, but I cannot see how it respects multiplication as $(bcc'a) \neq (bca)(bc'a).$ Any help is appreciated, thanks.