''x'' characterized named ''f'' applies topos whether concrete concrete case namely ''c'' faithful example category sets situation reduces familiar behavior functions monics ''m'' ''x '' rarr ''x'' exactly injections one one functions ''x '' ''x'' given image ''mx'' ''x'' ''x '' constitute subobject ''x'' corresponding morphism ''f'' ''x'' ''f'' sup minus sup ''t'' image monics subobject general many
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