2nd - Auxiliary

Auxiliary function by Daryl Sharp

A helpful second or third function, according to
Jung’s model of typology, that has a co-determining influence on
consciousness.

Absolute sovereignty always belongs, empirically, to one function alone, and can belong only to one function,
because the equally independent intervention of another function would
necessarily produce a different orientation which, partially at least, would
contradict the first. But since it is a vital condition for the conscious
process of adaptation always to have clear and unambiguous aims, the presence
of a second function of equal power is naturally ruled out. This other
function, therefore, can have only a secondary importance. . . . Its secondary
importance is due to the fact that it is not, like the primary function . . .
an absolutely reliable and decisive factor, but comes into play more as an
auxiliary or complementary function.[“General Description of the
Types,” CW 6, par. 667.]

 

The auxiliary function is always one whose nature differs from, but is not antagonistic to, the superior or primary
function: either of the irrational functions (intuition and sensation) can be
auxiliary to one of the rational functions (thinking and feeling), and vice
versa.

Thus thinking and intuition can readily pair, as can thinking and sensation,
since the nature of intuition and sensation is not fundamentally opposed to the
thinking function. Similarly, sensation can be bolstered by an auxiliary
function of thinking or feeling, feeling is aided by sensation or intuition,
and intuition goes well with feeling or thinking.

The resulting combinations present the familiar picture of, for instance, practical thinking
allied with sensation, speculative thinking forging ahead with intuition,
artistic intuition selecting and presenting its images with the help of
feeling-values, philosophical intuition systematizing its vision into
comprehensive thought by means of a powerful intellect, and so on.[Ibid., par.
669] ^[1]

                

source:

Jung Lexicon:
A Primer of Terms & Concepts by Daryl Sharp, M.A. Jungian Analyst

The Source used is from a website rather than directly from Beebe Book

Auxiliary function by John Beebe

The Auxiliary function, (also called the Parent Function) is our supporting function, it acts as a support to our dominant function and will always be an opposite orientation, meaning if your dominant function is introverted, your auxiliary function will be extroverted, if your dominant function is a judging function, your auxiliary function will be a perceiving function. It's called the "Parent" function as we typically use this function to help and compliment ourselves and others. 

According to Carl Jung, the auxiliary function starts to be developed during our teen years, we start developing this function in ages 13-20. (Note that this is only approximate, everyone matures differently)

Examples:

Te Auxiliary: Will support or back up the theory/intuitive perception of dominant Ni, or back up/support the personal experience/sensory perception of dominant Si using empirical facts. 

Fe auxiliary: Will support or back up the theory/intuition perception of dominant Ni, or back up/support the personal experience/sensory perception of dominant Si using community values.

Source: https://presentobsessions.wordpress.com/2013/12/03/the-john-beebe-8-function-model/

References

Source:[1]Jung Lexicon:
A Primer of Terms & Concepts by Daryl Sharp, M.A. Jungian Analyst