Models in categorical logic

Question

This is a very basic question: given a theory $\mathbb{T}$, I have seen definitions of models of $\mathbb{T}$ as functions from signature $\Sigma$ to a fixed background category such that it satisfies all sentences in $\mathbb{T}$. On the other hand I have also encountered mentioning of models as objects in that category. It is not obvious to me that the two definitions are compatible with each other, but perhaps I am missing something obvious.

Answer 1

Absolutely not. To be a model is all about the interpretation of formulas. What should it mean that $M \models \phi$?

Joyal gave a very good answer to this question in Elementary toposes. Something is done in general cartesian categories.

You can find this in many books, from Sketches of an Elephant (pag 800-900), or Sheaves in Geometry and Logic.