The following is a theorem in Differential Equatios book by Boyce & DiPrima:
If $F(s) = \mathcal L\{ f (t)\}$ exists for $s > a \ge 0$, and if $c$ is a positive constant, then
$\mathcal L\{u_c(t)f(t − c)\} = e^{−cs}\mathcal L\{ f (t)\} = e^{−cs}F(s), s > a$.
Conversely, if $f (t) = \mathcal L^{−1}\{F(s)\}$, then $u_c(t)=f(t − c) = \mathcal L^{−1}\{e^{−cs}F(s)\}$.
I just didn't understand the condition $s>a \ge 0$. Is positiveness important here?