Function spectrum with values in a ring spectrum
= Function spectrum with values in a ring spectrum
If $E$ is a <ring spectrum>, then $F(X_+,E)$ is a <ring spectrum>. Its multiplication is adjoint to evaluating the two functions on the two copies supplied by the diagonal of $X$, then multiplying in $E$. Its unit is the constant unit function. Coassociativity of the diagonal and associativity in $E$ prove the associative identity; the counit and unit identities prove both unit laws.