I have a problem here with this statement in Chapter 2,Details:
In particular, $T$ is realized if and only if its identity functor is a model.
Namely, I do not know how the identity functor arises and between which categories it leads ?
Is it $id:T\to T$ or $id:C\to C$ ?
How do follow both directions of the above iff ?
$T$ is realised if and only if the identity functor on the category $T$ is a model. If $\mathrm{Id}_T : T \to T$ is a model, then each (co)cone in $T$ is sent to a (co)limit (co)cone in $T$, which means each (co)cone must have been (co)limiting in the first place. Conversely, if each (co)cone in $T$ is (co)limiting, then the identity will send each (co)cone to a (co)limiting (co)cone by definition, which means $\mathrm{Id}_T$ is a model.