Laplace transform of $u(t)=1/s$

Question

I know that the Laplace transform of $u(t)$ is equal to $1/s$ (causal system). But the Laplace transform of the impulse response of the integration operation is also equal to $1/s$. Intuitively, could someone tell me how they are related? $u(t)$ is a constant for $t>0$. But an integrator's output is increasing linearly with respect to time.

Answer 1

I suppose I will answer my own question here but please do add additional comment if there is a better way to interpret it. So can we say that the inverse Laplace transform of 1/s is u(t) which exactly describes the integration operation because integration keeps its output forever? That is why it is a constant when we put an impulse into the system. The output, u(t), describes what the output looks like after a spike is applied to the input. It just remembers it.