Sigue la camara al personaje manteniendo siempre un desplazamiento con respecto a donde está avanzando

CameraFollower.cs

using UnityEngine;
using System.Collections;

public class CameraFollower : MonoBehaviour
{

    public Transform target;
    public float initOffsetX;
    public float initOffsetY;
    public float moveToRightOffset;
    public float moveToLeftOffset;
    public float moveToBottomOffset;
    public float moveOffset;

    public float minCamaraPositionX;
    public float minCamaraPositionY;

    private float xOffset;

    private float yOffset;

    private void Awake()
    {
        this.xOffset = initOffsetX;
        this.yOffset = initOffsetY;
    }

    void Update()
    {
        float newPositionX = target.position.x + xOffset;
        float newPositionY = target.position.y + yOffset;
        if (newPositionX < minCamaraPositionX)
            newPositionX = minCamaraPositionX;

        if (newPositionY < minCamaraPositionY)
            newPositionY = minCamaraPositionY;

        transform.position = new Vector3(newPositionX, newPositionY, transform.position.z);
    }

    public void FollowUpOffset()
    {
        if (moveToBottomOffset < this.yOffset)
        {
            this.xOffset -= moveOffset;
            if (this.yOffset < moveToBottomOffset)
                this.yOffset = moveToBottomOffset;
        }
    }

    public void MoveToLeftOffset()
    {
        if (moveToLeftOffset < this.xOffset)
        {
            this.xOffset -= moveOffset;
            if (this.xOffset < moveToLeftOffset)
                this.xOffset = moveToLeftOffset;
        }
    }

    public void MoveToRightOffset()
    {
        if (moveToRightOffset > this.xOffset)
        {
            this.xOffset += moveOffset;
            if (this.xOffset > moveToRightOffset)
                this.xOffset = moveToRightOffset;
        }
    }
}