Nietzsche's role as a philosopher is that he reintroduces the importance of subjectivity back into the intellectual realm, where it had, historically and culturally, lost much of its status. Yet he does not want to overturn the intellect with his approach. That is a common misunderstanding. Rather, he is seeking to give expression to the inner self, a self that is constructed along lines of will to power. *    Zarathustra's "bridge" is a metaphor for crossing from everyday normality to transcendence of one's fearful or cowardly states.

To attempt to solve the problem of the lack of magnificence in humanity by immersing oneself in either subjectivity or objectivity alone is wrong. In Nietzsche's terms, this would imply a form of cowardice and moral failing in not living up to one's human potential. Both of these would-be solutions are in error. The abyss between normative being and Overman has to be crossed by individuals representing humanity.

The metaphor of the "bridge" in Nietzsche's Zarathustra is quintessentially shamanistic, although it actually alludes to the evolutionary development of humankind beyond the apelike state: a process that is ungoing and far from finished.  Within a work of art, there are many levels of resonance, however, and given the way that Nietzsche focuses so much on the untapped resources of instinct, emotion, a feeling for the past, and bhuried treasures within the psyche, I think  that the bridge can also be understood to access the resource of one's subjective states from the standpoint of intellectual objectivity – moving across and downwards into the structures of the unconscious. To cross the bridge "as spirit" is a term for becoming shamanised according to an expert in the field, Mircea Eliade. Nietzsche picks up the shamanistic theme and introduces into it the notion of the Superman. From Zarathustra:

I love him who reserveth no share of spirit for himself, but wanteth to be wholly the spirit of his virtue: thus walketh he as spirit over the bridge. [note the shamanistic trope]
I love him whose soul is deep even in the wounding, and may succumb through a small matter: thus goeth he willingly over the bridge. [note the reference to shamanistic wounding, leading to shamanistic actualisation]
I go not your way, ye despisers of the body! Ye are no bridges for me to the Superman!—[note the shamanistic link to the body as the means by which a bridge is forged to the 'spirit' realm.]

It is quintessentially shamanistic to build a bridge over an abyss of consciousness, thus uniting aspects of the mind that were before separated. I suggest that this abyss may be most clearly understood as existing between two parts of the mind – the neo-cortex and R-complex – although an experientially founded and existential rendering of the shamanistic pattern can open it up to some idiosyncratic depictions.

Due to the estranged nature of current human consciousness from itself, it is imperative, as part of the project of developing human consciousness, to make an inwards, and experiential bridge leading back from the realm of factual knowledge (the realm instrumental reason and consciousness) to the subjective realm of knowledge (the unconscious and its will to power). On the most significant level, the two levels of knowledge represent a structural divide between consciousness and the Nietzschean unconscious (unexpressed potential,  oriented to the world as will to power.)

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* Kleinian "object relations" literature would revise the element of emotion that Nietzsche sees as the individual's abundance within a framework of scarcity, neediness and survival, which she alleges to be a key element of the early, most subjective levels of consciousness. But Nietzsche's concept is that the unconscious has something more life enhancing and actively aggressive.