A ceremony for the first woman in one of the Church's senior positions was held at Chester Cathedral on International Women's Day.
Bishop Libby, 48, was consecrated as the eighth Bishop of Stockport at York Minster in January but yesterday's event was the start of her ministry.
She is the first female to become a bishop after the Church passed legislation last November following decades of debate.
Chester Cathedral was packed with more than 1,900 people from across the diocese.
She said the service was a "homecoming" and that she continues to "feel deeply grateful for the honour of this calling and the privilege of exercising it in this place".
She added: "Expectations are high, and I too am excited by the possibilities and challenges ahead. I echo what I said in my statement at my announcement - the church faces wonderful opportunities, to proclaim afresh, in this generation, the good news of Jesus and to build his kingdom.
"I am delighted to be engaging with that task in an area I know and love and among people I value so highly. But I do know I'll need people to be patient and understanding - about the time it will take for me to address all that is already waiting for me, and for the mistakes I will inevitably make on the way."