Obtaining Definition of Coalgebra via Principle of Duality in Category Theory

Question

I have been reading about this principle of duality in category theory. Although I understand the main idea, I am curious to know how one can formalize this. The form of duality I have been given is

Theorem (Duality). For every Well-Formed Formula $\phi$ in the language of category theory, $\mathsf{dual}(\phi)$ holds in category $\mathfrak{C}$ if and only if $\phi$ holds in the opposite category $\mathfrak{C}^{\mathsf{op}}$.

As an exmaple, I tried to obtain the definition of an $F$-coalgebra from that of an $F$-algebta. For this, let us consider the definitions of $F$-algebra and $F$-coalgebra in category theory

Definition 1 ($F$-Algebra). Suppose $F: \mathfrak{C} \to \mathfrak{C}$ is a functor over the category $\mathfrak{C}$. An $F$-algebra is a pair $(A, \alpha)$ where $A \in \mathsf{obj}(\mathfrak{C})$ and $\alpha: F(A) \to_{\mathfrak{C}} A$.

Definition 2 ($F$-Coalgebra). Suppose $F: \mathfrak{C} \to \mathfrak{C}$ is a functor over the category $\mathfrak{C}$. An $F$-coalgebra is a pair $(A, \alpha)$ where $A \in \mathsf{obj}(\mathfrak{C})$ and $\alpha: A \to_{\mathfrak{C}} F(A)$.

Here are my questions.

1. What is this language of category theory? Is this referring to a formal language? If YES, is it a first order language? How $\mathsf{dual}$ is defined in such a context?

2. What does it mean for a WFF to hold in a category? Does it mean that there is a model which is somehow related to this category? Can I view the theorem as something like

Theorem. For every $\phi \in \mathsf{FRM}_{\mathfrak{L}}$, $\mathfrak{C} \vDash \mathsf{dual}(\phi)$ if and only if $\mathfrak{C}^{\mathsf{op}} \vDash \phi$.

3. How should I derive Definition 2 from Definition 1 in a rigorous way using the above theorem? So, I am looking for a mechanical way as above theorem suggest to show that