"조련사는 민지라고 합니다."
Translation:The trainer is named Minji.
Yes, so this is quoting with 이다: ~(이)라고. When the noun before 이다 ends with a vowel, "이" can be omitted.. so for example, both of these would be correct:
- 그가 의사이라고 했어요. (O)
- 그가 의사라고 했어요. (O)
But if the noun before 이다 ends with a consonant, "이" cannot be omitted:
- 그가 선생님이라고 했어요. (O)
- 그가 선생님라고 했어요. (X)
though koreans tend to say 의사라고 했어요, rather than 의사이라고 했어요. This is something that when something you want to say doesnt have a consonant beneath the last syllable, the postponents would be 를or 가. if it does, it would be 을, or 이. this sentence would follow the exact rule, hence that is why the 선생님 has 이라고, and 의사 has 라고 next to them.