Relationship with Ofelia
According to Guillermo del Toro, the Faun is "a creature that is neither good or evil....like nature....a character there to be witness and shephard her (Ofelia) in her rite of passage, but he has no agenda. He doesn't care if she dies or lives." In spite of this, it is strongly implied that the Faun lusts after or loves Ofelia/Moanna.
After Ofelia completes the first task, she reunites with Faun inside the labyrinth, where she notices a statue of a Faun, a woman, and a baby. The Faun explains that the statue is of him and her, and although it is not stated explicitly, we can assume that the baby is theirs. If this is true, then one of two theories are correct. Either the Faun and Moanna were lovers or spouses in the Underworld and raised a child, or the Faun can predict the future and knows that he is destined to be Moanna's husband and father of her child.
Ofelia's feelings towards the Faun are complicated. She wants to trust him and often views him as a source of comfort, like the father figure that the ruthless Captain will never be. After completing the first task, she grows uncomfortable with him as he strokes her face and tells her that they will soon be reunited and walk through the seven gardens of her palace. She asks him, "How do I know you speak the truth?" As much as she wants to believe him, she questions his motives. This and the Faun's indifference towards Carmen's health lead to her disobeying him as she carries out the next two tasks.