Agni mukham is exactly the end of poorvanga prayoga and the start of uttharanga prayoga. Poorvanga prayoga just simply means pre-procedure. Poorvanga prayoga includes kalasa sthabanam or invocating the diety in a water pot, yantra sthabana or construction of yantra, consecrating fire altairs according to the procedures, doing the necessary mantra invocations and setting the ambience for the fire rituals. All these fall under poorvanga prayoga. Uttharanga prayoga is anything done in core with fire or after the consecration process.
Legends has that Lord Agni was appointed as a bridge between the worlds and directly blessed by lord vishnu. Lord Agni is described as two faces with seven tongues and seated on a mesha vahana or aries chariot. It is since time immemmorial yagyams are considered to be the greatest wealth and the achievement one can cherish about. Best is eg is our former prime minister Vajpai, the suffix Vajpai is due to the vajpayee yagya done by his ancestors which became the suffix of their lineage such is the credit of a yagyam. Yagyam or fire rituals, is the only sort after solution to any problem. Agni mukham is performed once the poorvanga yagyam is completed. The fire Altair is then purified with necessary mantra invocatoons and the first reverence is paid to Sri Varaha Swamy as he is said to be the form of altair. Then lord fire is invoked with his consorts in the altair fire. Lord Agni has two consorts Swaha Devi and Shwatha Devi. Whenever an yagyam is done for the sake of deities Swaha Devi is invoked. Whenever there is a fire ritual for Ancestor related rituals Shwatha Devi is invoked. It is only after the Agni Mukham the things scheduled to offer is purified in the same fire ignited.
By the fire sacrifice that I perform.
Let the fire sacrifice give me long life,
Let fire sacrifice give me inhaled air,
Let fire sacrifice give me exhaled air,
Let fire sacrifice give me able eyes,
Let fire sacrifice give me able ears,
Let fire sacrifice give me able mind,
Let fire sacrifice give me able words,
Let fire sacrifice give me an able soul,
Let fire sacrifice give me ability to perform more fire sacrifices. ( chamakam )