For a simple ODE: $u' = u, \: \: \: t \ge 0$, with initial condition $u(0)=1$, the exact solution is $$u(t) = e^{t}$$
Now if the initial condition $u(0) = u_{0}$ then the exact solution is written $$u(t) = u_{0}e^{t}$$
Now define $\phi(t; u_{0})$ as the exact solution of the ODE above with initial condition $u_{0}$. For example: $$\phi(t; 1) = e^{t}$$ $$\phi(t; -5.9) = -5.9e^{t}$$ $$\phi(t; 0.11) = 0.11e^{t}$$
This definition is useful as follows: $$\phi(0.5; \phi(0.5; 0.11)) = \phi(1; 0.11) $$
but cumbersome to read.
So instead of defining $\phi$ as a function with parameter of initial condition, how can I properly/rigourously define $\phi$ as a time-dependent operator such that
$$ \phi(t)(u_{0}) = u_{0}e^{t}$$ so that I can neatly write, for example
$$ \phi^{2}(0.1) u_{0} = \phi(0.1)\phi(0.1)u_{0} = \phi(0.2) u_{0} $$
It is quite standard to write $\phi_t(x_0)$ for $\phi(t;x_0)$, sometimes also in the superscript position. Then the (semi-)group property for the flow of autonomous DE can be conveniently written as $$\phi_t\circ\phi_s=\phi_{t+s}.$$
In the superscript variant one would get the suggestive $(\phi^t)^{\circ n}=\phi^{nt}$.