Персонаж должен был делать флип по оси x

using UnityEngine;
using System.Collections;
 
public class Character : Unit
{
    [SerializeField]
    private int lives = 5;
 
    
 
    [SerializeField]
    private float speed = 3.0F;
    [SerializeField]
    private float jumpForce = 15.0F;
 
    private bool isGrounded = false;
    [SerializeField]
    private Bullet bullet;
 
    private CharState State
    {
        get { return (CharState)animator.GetInteger("State"); }
        set { animator.SetInteger("State", (int)value); }
    }
 
    new private Rigidbody2D rigidbody;
    private Animator animator;
    private SpriteRenderer sprite;
 
    public void Awake()
    {
        
        rigidbody = GetComponent<Rigidbody2D>();
        animator = GetComponent<Animator>();
        sprite = GetComponentInChildren<SpriteRenderer>();
 
        bullet = Resources.Load<Bullet>("Bullet");
    }
 
    private void FixedUpdate()
    {
        CheckGround();
    }
 
    private void Update()
    {
        if (isGrounded) State = CharState.Idle;
 
        if (Input.GetButtonDown("Fire1")) Shoot();
        if (Input.GetButton("Horizontal")) Run();
        if (isGrounded && Input.GetButtonDown("Jump")) Jump();
    }
 
    private void Run()
    {
        Vector3 direction = transform.right * Input.GetAxis("Horizontal");
 
        transform.position = Vector3.MoveTowards(transform.position, transform.position + direction, speed * Time.deltaTime);
 
        sprite.flipX = direction.x < 0.0F;
 
        if (isGrounded) State = CharState.Jump;
    }
 
    private void Jump()
    {
        rigidbody.AddForce(transform.up * jumpForce, ForceMode2D.Impulse);
        if (isGrounded) State = CharState.Run;
    }
    
    private void Shoot()
    {
        Vector3 position = transform.position; position.y += 0.8F ;  position.x += 0.5F;
        Bullet newBullet = Instantiate(bullet, position, bullet.transform.rotation) as Bullet;
 
        
    newBullet.Direction = newBullet.transform.right * (sprite.flipX ? -1.0F : 1.0F);
    }
 
  
    private void CheckGround()
    {
        Collider2D[] colliders = Physics2D.OverlapCircleAll(transform.position, 0.3F);
 
        isGrounded = colliders.Length > 1;
 
        if (!isGrounded) State = CharState.Jump;
    }
 
   
}
 
 
public enum CharState
{
    Idle,
    Run,
    Jump
}

newBullet.GetComponent<Sprite>().flipX = sprite.flipX;

private void Shoot()
    {
        Vector2 position = transform.position; position.y += 0.8F ;  position.x += 0.5F;
        Bullet newBullet = Instantiate(bullet, position, bullet.transform.rotation) as Bullet;
        newBullet.Direction = newBullet.transform.right * (sprite.flipX ? -1.0F : 1.0F);
        newBullet.GetComponent<SpriteRenderer>().flipX = sprite.flipX;
 
    }

position.x += sprite.flipX ? 0.5F : -0.5F;
// Или так:
position.x += sprite.flipX ? -0.5F : 0.5F;
// В зависимости как у тебя изначально спрайт повернут.