hi,
usually it doesn't work this way
ic's like lm393n can be tested on output side with diode tester, they usually have shorted output transistor, and in his datasheet in the simplified schematic you see some npn or pnp transistor as output transistor. for the input side, it's not the same, as the inputs are more 'complicated' schematics.
so what we can do? for AC, use the scope and see if ac waveform is amplified as it should be, for DC we can verify if he 'triggers' in the right direction. that means we understand very well how opamp works.
that 'shorted output' diagnose works from my experience in maybe less than 50% situations, usually when the designer left the possibility of opamp output to be overloaded on output side (like not protecting him from oscillations, short circuits - i found opamps with short-circuit protection included to be shorted!, bad layout on output side etcaetera). inputs can suffer the same and you got the input protection diodes or input transistors shorted or smoked, in that case you won't see the short with your diode tester.
finally, as @james_s pointed, the test schematic from datasheet is finally the 100% reliable test, but anyway often enough we find that one implemented with +- modifications on our defective board