= Solution
For $K\leq_m\mathbf{Cof}$, map $x$ to a machine that initially enumerates nothing and, if the simulation establishing $x\in K$ halts, enumerates every word. Its language is cofinite exactly when $x\in K$.
For $\mathbb B\setminus K\leq_m\mathbf{Cof}$, map $x$ to a machine that enumerates successively longer finite initial segments of $\mathbb B$ while simulating $x$. If the simulation never halts, every word is eventually enumerated; if it halts, enumeration stops with a finite language and therefore an infinite complement. Thus the constructed language is cofinite exactly when $x\notin K$.
The parameter theorem makes both code transformations total and computable.
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