The phrase "gat into head" is an informal term used to describe something that has been firmly understood or has deeply penetrated one's mind. It can be replaced with a variety of synonyms, such as ingrained, impressed, absorbed, entrenched, or imprinted. These words suggest that the idea has become a part of one's belief system or way of thinking, indicating a strong and lasting impression. Other synonyms include infixed, rooted, indelibly marked, deeply seated, and firmly established, which all convey the idea of a resilient and unyielding idea or belief that refuses to be shaken or displaced from the mind.