First the ECM has to get the engine to start, add fuel to cold engine till it fires. The ECM is looking for the range that fuel will burn, in the cylinder.
Air/Fuel Ratio Limits 6.0:1.......Rich run limit
9.0:1...... Low power, black smoke
11.5:1.... Rich best torque at WOT
12.5:1.... Safe best power at WOT
13.2:1.... Lean best torque at WOT
14.7:1.... Chemically ideal
15.5:1.... Lean light load, part throttle
16.2:1.....Best economy, part throttle
18-22:1.. Lean run limit
Somewhere between 22:1 and 6:1 (in the cylinder) the engine will "catch/fire". This is after all the fuel, that has to be loaded to "wet the walls" (the total surface area from the injector to the combustion chamber). If carb/TBI read lots of surface area.
If the cylinder does not "catch/fire", then it will have to wait till the next pass/cycle, pushing all that mixture into the exhaust.
Hopefully if all goes well (engineer or you being the engineer/programmer, loaded correct values into ECM ) then the engine will start and run.
Immediately after firing something happens called heat. As things warm up, all the "wet the walls" fuel that was needed to start the engine is now being added to the mixture. This will require that the amount of fuel being added by the ECM to be decreased to maintain the burnable AFR in the cylinder.
At the same time all this, "must be done this way to start engine stuff" is happening, the exhaust and cat is catching and trying to burn all the excess.
When everything is nice and hot the engine reaches a steady state of operation. The ECM will add "X" amount of fuel to "Y" amount of air (see chart above) and down the road you go.
Now comes the EPA and the Hydrocarbon (unburnt fuel) emissions amount. This has to be dealt with, so they added a cat converter, to burn the extra fuel. A cat can not run at the same steady state like the engine, it needs to run at it's own correct AFR (low HC output). To solve this dilemma, the O2 monitors the gases between the engine and the cat, but it can not serve two masters.
What the converter needs, to maintain a steady burn (low HC output), has to be delivered through and after the engine, like making a meal from leftovers.
So the ECM plays with (adds/removes fuel) to the cylinder AFR to satisfy the converter AFR. If and when the engine needs more power (richer AFR), also adding more air to manifold/cat via the air pump.
The game the ECM/O2 play is a up and down (richer/leaner) game, all in the name of lower HC. A cold converter will not burn, so some fuel is deliberately wasted to keep the cat hot (at a self burn temp). Too much air/fuel volume and the converter will melt, too little and the fire goes out.
This why open loop tuning (no cat) is attractive, it is all "steady state tuning".
Closed loop is anything but steady. So when you are being the "engineer/programmer", the more you know about how things work and when, the easier and "more correct" your adjustments will be.
The O2 makes adjustments at a lot higher rate then the sampling rate of the ECM, so all you see is, the strobe light effect of the ECM reading, on the O2 output, not what the AFR is really doing.
Some WB's can see the lean/rich/lean of a single combustion pulse. Remember that 4 cycle engines are all about pulses, rich/lean per strokes, not a steady flame type of burn (furnace/stove/lighter).
Engine/cat AFR adjustments are made long after the fact (time lag), from when the actual burn took place, (down the pipe is way later, "headers").
Tail pipe WB readings are more "averaged" due to mixing of the gases in the cats and/or exhaust pipes, then the usual NB reading at the manifold.
The new stuff has right side, left side, pre, and after cat O2's