"Many things including such matters as parent's character, his disposition, his emotional stability, his neglect or indifference toward child, his trustworthiness, and his acts and conduct may have bearing on parent's fitness, and may be considered by trial judge in deciding question of child's custody." In re Wisdom (1956) 146 Cal.App.2d 635.
Evid. Code 1100. "Except as otherwise provided by statute, any otherwise admissible evidence (including evidence in the form of an opinion, evidence of reputation, and evidence of specific instances of such person's conduct) is admissible to prove a person's character or a trait of his character."
Evid. Code 1101. "(a) Except as provided in this section and in Sections 1102, 1103, 1108, and 1109, evidence of a person's character or a trait of his or her character (whether in the form of an opinion, evidence of reputation, or evidence of specific instances of his or her conduct) is inadmissible when offered to prove his or her conduct on a specified occasion.
(b) Nothing in this section prohibits the admission of evidence that a person committed a crime, civil wrong, or other act when relevant to prove some fact (such as motive, opportunity, intent, preparation, plan, knowledge, identity, absence of mistake or accident, or whether a defendant in a prosecution for an unlawful sexual act or attempted unlawful sexual act did not reasonably and in good faith believe that the victim consented) other than his or her disposition to commit such an act.
(c) Nothing in this section affects the admissibility of evidence offered to support or attack the credibility of a witness."
Evid. Code 1104. Except as provided in Sections 1102 and 1103, evidence of a trait of a person's character with respect to care or skill is inadmissible to prove the quality of his conduct on a specified occasion.
From the above, it appears that character evidence in a child custody action is admissible based upon case law precedent, but not based the actual Evidence Code. Until you asked the question, I hadn't reviewed the specific laws -- but it does seem that the entire Family Court
Judicial System seems to be operating under a misunderstanding that the Evidence Code supports the use of character evidence for purposes other than impeachment (Section 1101(c)), as part of a custody action. In particular Section 1104 appears to completely exclude character as a means of proving that your skill as a parent is deficient, based upon any witness' opinion (including your child's) of your general demeanor.
So, you can object if the child is asked questions about your general conduct, as inadmissible under Evidence Code 1104. Of course, testimony of specific incidences of your conduct on prior occasions is admissible, because it is not character evidence.
Hope this helps.