The dual imagery of fish and sheep in John 21 highlights both the evangelistic and pastoral mission of the church ("pastoral" involves care for sheep). When the church casts the Gospel net and draws in fish, the fish become sheep in the flock. Caring for sheep is not a ministry limited to official shepherds. The shepherds are "overseers" of the flock (1 Tim 3.1), which means they have the responsibility to ensure the sheep are safe and healthy. However, the day to day care of the sheep is not exclusively their ministry. The New Testament bears this out with some instructions and examples.
In Acts 6, some Grecian widows were overlooked in the daily distribution of food. Without a welfare system in the way we usually think of it, widows were dependent on their families. In the case of Christian widows with no family, the church was their family. With a God-given mandate to care for its own, the church had a "meals on wheels" program for its destitute widows. At this time, the church was still heavily Jewish, which may have been why some of the gentile widows were overlooked. When this came to the attention of the leaders, they took responsibility to ensure these widows were cared for by charging the church to choose trusted servants to take charge of the day to day operation of this ministry whom the leaders could appoint. It is interesting to note that one of the words for servant was "diacanos," from which we get the word "deacon." Deacons in the early church cared for and served those in need in the church.
Using the body as an analogy for the church, 1 Corinthians 12 teaches all parts of the body are interconnected and should have the same care for one another. Caring for one another is a mission of the church and not the exclusive ministry of shepherds. The church is not merely a flock of helpless sheep, but also a body in which all the parts care for each other.
In Matthew 25, Jesus said we are to feed, clothe, visit in prison, and generally care for "the least of these brothers of mine," which indicates he is talking specifically about caring for the brethren, the body of Christ. Therefore, when we feed, clothe and care for the body, we are doing so to him. This is a clear mandate for each part of the body of Christ to care for one another. This is for the church, not just a select few.
Add to these examples the many "one another" passages in the New Testament, and you have a strong mandate for the pastoral mission of the church. So, while we are to do good to all men, we are to do so especially for the household of faith. Caring for the flock does not belong exclusively to the elders and deacons, but to the church as whole.
So, an integral part of the fishing ministry of the church is the pastoral ministry of the church. After fish become sheep, they are integrated into the body and become part of the caring ministry of the church where they are both cared for and care for.