Now while Peter was inwardly perplexed as to what the vision
which he had seen might mean, behold, the men that were sent by Cornelius,
having located Simon's house, stood at the gate and called out ask if Simon called
Peter was lodging there. And while Peter was pondering the vision, the Spirit
said to him, "Behold, three men are looking for you. Rise and go down, and
accompany them without hesitation; for I have sent them." And Peter went
down to the men and said, "I am the one you are looking for; what is the
reason for your coming?" And they said, "Cornelius, a centurion, an
upright and God-fearing man, who is well spoken of by the whole Jewish nation,
was directed by a holy angel to send for you to come to his house, and to hear
what you have to say." So he called them in to be his guests.
Peter was staying with another Simon, a tanner who owned a house near the coast. Peter had a place of honour there. Some scholars say that Luke mentions the occupation of Peter's host to emphasise that Peter was already willing to have contact with a Jew who had an 'unclean' occupation. If so, Peter was already able to mix as his master had done when He ate with tax collectors and sinners. Peter could hardly have forgotten the persistent criticism that had come from over-scrupulous religious leaders. But I suppose we are being led along by Luke, because it's about Peter being led by the Holy Spirit one step at a time to where we are now. Will he be able to openly and without hesitation have contact with gentiles? Will he shake hands with them? Will he be able to be warm-heartedly hospitable to Cornelius' messengers?
Remember the occasion when the centurion at Capernaum had a servant on the point of death? Luke told us about this in his gospel (7:1-10). On that occasion the centurion had sent Jewish elders to Jesus to solicit His aid. Jesus went and near the house the Centurion sent others saying, in effect, "You don't have to break your customs for me!" and Jesus took this as a miracle of faith and the servant was healed without Him entering the house of that well-loved Gentile.
Luke doesn't tell us if Cornelius had sent Jewish servants to Peter. And neither did the Holy Spirit who spoke to Peter before introductions were made. Peter could be in no doubt as to what his response had to be. He was to "accompany them without any hesitation" and then the Spirit of Jesus added "for I have sent them." Not Cornelius. The Holy Spirit told Peter that these were His messengers. Luke tells us that Peter obeyed. He invited them in to be his guests. This says something about the respect in which Peter was held in that household. It also tells us that Peter realised his everyday conduct involved a transformation to some age-old customs. The Holy Spirit was prodding and prompting Peter to a wider vision, a vision that encompassed the whole earth. Cornelius' vision of the angel and Peter's strange dreams are now inseparable parts of the same story.