But she said, “Yes, Lord, for even the little dogs eat of the crumbs that fall from their lords’ table.” Bible

“for even...” It would not be correct to translate this as “but,” “however,” etc. The woman is not opposing Jesus in any way. She is merely pointing out that the little dogs do get crumbs when the family eats.

Also see commentary on Matt 15:26.

Commentary for: Matthew 15:27