This was my solution, if you use gameObject and gameObject.name, you seem to get the same response. I went with .name because it seemed more correct.
private void ShowNextHitSprite()
{
int spriteIndex = timesHit - 1;
if (hitSprites[spriteIndex] != null)
{
GetComponent<SpriteRenderer>().sprite = hitSprites[spriteIndex];
}
else
{
Debug.LogError("Block sprite is missing from array inside of - " + gameObject.name);
}
}