The correct answer is Option 2: Jahangir.

The English East India Company established its first factory in India during the time of Emperor Jahangir. Jahangir was the fourth Mughal emperor and ruled from 1605 to 1627. During his reign, the English East India Company, which was established in 1600, sought permission to establish a factory and trade in the Mughal territories.

In 1608, Jahangir granted a firman (royal decree) to the English East India Company, allowing them to establish a factory in the port city of Surat in Gujarat. This marked the beginning of the English East India Company`s presence in India and laid the foundation for their future influence and control over trade in the region.

The establishment of the factory in Surat was a significant milestone for the English East India Company. It provided them with a base from which they could conduct trade, primarily in spices, textiles, and other valuable goods. Over time, the English East India Company expanded its operations in India and gradually gained more influence, leading to the establishment of more factories and the eventual colonization of large parts of the country.

It is worth noting that during the reign of Jahangir, the Mughal Empire was at its peak in terms of territorial extent and cultural influence. The Mughal emperors, including Jahangir, initially viewed the European trading companies, including the English East India Company, as potential allies and sources of revenue. They granted them trading rights and allowed them to establish factories to strengthen economic ties.

However, over time, the East India Company`s growing influence and the expansion of their territorial control led to conflicts with the Mughal Empire, particularly in the later years of the Mughal dynasty during the reign of Aurangzeb. This eventually paved the way for the British colonization of India.

In conclusion, the English East India Company established its first factory in India during the time of Emperor Jahangir, the fourth Mughal emperor. The grant of a firman by Jahangir allowed the Company to establish a factory in Surat, marking the beginning of their presence in India and setting the stage for their future dominance in the region.