A relatively easy source on that is Unique Tenets of The Middle Way Consequence School, by Daniel Cozort. Better ones are Khedrup Je's Dose of Emptiness, and Shantideva's Bodhisattva Deeds.
Vaïbashikas and Madhyamika-Prasangika do not assert self-knowers (self-cognizers). The other schools do. However, if Prasangika refute self-knowers, they do assert that awareness is self-knowing. It just does not know itself in the way that other schools posit. Prasangika refute self-knower as presented by other schools. Other schools posit a self-knower that is:
- special in that it is neither a main mind nor a mental factor
- realizes an apprehender (i.e. another consciousness) in a direct and non-dualistic way
- one entity with the consciousness it knows.
The idea is that other schools of thoughts (tenets) from Svatantrikas down to and including Sautrantikas claim that if there was no self-knower, one could remember blue but could not remember having seen blue. Prasanngika claim that there is no need for self-knowers to be able to remember having seen blue, because the eye-consciousness apprehending blue implicitly realized itself at the time of seeing blue. Implicitly means by way of the object not appearing. So, when one remembers blue, he also remembers "having seen blue" by way of relation; realizing having seen blue is induced by remembering the object, because the consciousness and its object were already related by the time of seeing.
What follows are arguments from Geshe Gyaltsen's teachings. There are a commentary to the chapter 9 of Shantideva's Bodhisattva Deeds.
Prasangika: "There is no self-knower realizing an apprehender directly
and explicitly in a non-dual manner, because Buddha said mind cannot
see itself as the sword cannot cut itself."
Prasangika: "If a self-knower was to realize itself in a non-dualistic
manner, it would follow that [the apprehended object and its
apprehending consciousness] are of the same isolates."
Cittamatrin: "The self-knower is like a butter lamp, illuminating both
the outside and itself."
Prasangika: "The butter lamp does not illuminates itself since it does
not need to be illuminated. The butter lamp does not need to
illuminate itself since it does not obscure itself[...] The fact that
the consciousness illuminates is known by the consciousness, which is
of a different substance. 'Consciousness illuminates' is known by
what? By another consciousness or by itself? If we say consciousness
illuminates and that is known by another consciousness, that
consciousness should be known again by another consciousness and that
should go on like this without end."
Prasangika: We do not need that self-knower since the eye
consciousness explicitly realizes its object and implicitly realizes
itself, while realizing its object.
Through relation of experiencing another, that is when the eye
consciousness sees blue, it also experiences itself, in relation to
realizing blue. Therefore there is no point in asserting a
self-knower.
This is possible because both do not exist inherently and therefore
can be related. By remembering the object, one implicitly realizes the
subject.