A lion is a cat - Lew TO kot [lew (nominative case) = kot (nominative)]
The sentence "Lew TO kot" means: "A lion" = "a cat" (the same thing)
A lion IS a cat - Lew JEST kotem [lew (nom) JEST kotem (instrumental)]
The sentence "Lew JEST kotem" uses the verb JEST to define, to describe
the lion, and requires that descriptive noun be in the instrumental case.
Lew JEST psem (instrumental)
Lew nie JEST psem (instrumental)
to be - być
it is - JEST
Instrumental - Narzędnik (kim? czym?) JEST psem/Nie JEST psem
This case is used to define, or to describe thing(s) in a full sentence.
The verb JEST puts the noun in direct object and needs the instrumental:
Lew (nominative) JEST psem (instrumental) - A lion IS a dog
Lew (nominative) nie JEST psem (instrumental) - A lion IS not a dog
In the phrase: "Lew TO (jest) pies" - "jest" goes to the back of your head.
The "TO" gives enough information: a lion (subject) = a dog (same thing)
Therefore, "TO" behaves like the English predicate nominative, which refers
back to the subject (nominative) after the verbs "to be" and "to become":
Lew TO pies (nominative) - A lion (nominative) is a dog (nominative)
Lew TO nie pies (nominative) - A lion (nominative) is not a dog (nominative)