Let $M$ be a ordinary Turing machine. Intuitively, we can say that $M$ can be transformed to a machine $M_D$ with doubly infinite tape, by placing to the left on every input $w = w_1, \dots , w_n$ some symbol $\#$ that tells the machine to 'get back' to the right cell.
So rigorously what does this entail?
If we $q$ is the start state of $M$, can we define $q'$ to be the start state of $M_D$ so that $\delta(q', w_1) = (q', w_1, L)$ and $\delta(q', \sqcup) = (q, \#, R)$ ? Then we can define $\delta(x, \#) = (x, \#, R)$ for any state $x$.
Am I thinking about this too hard, or is there an easier way to go about this?