Existence of Spec functor

Question

I want to prove the theorem A.2 of this (page 213).

Theorem: Given a proper left exact category $\mathcal{E}$. Then, to any $\mathbb{T}$-algebra $R:\mathbb{T}\rightarrow \mathcal{E}$, there exists a proper left exact functor $\overline{R}:FG\mathbb{T}^{\text{op}}\rightarrow \mathcal{E}$ such that $R=\overline{R}\circ F$, where $F(n)$ is the free algebra of $n$ generators.

The following is what I think:

1. Let $A$ be a finitely generated $\mathbb{T}$-algebra. $A$ is a colimit of finitely presented algebras $\{A_i\}$.
2. Each $A_i$ is a coequalizer of free objects.
3. Thus $\overline{R}(A_i^{\text{op}})$ is determined by $R$ as a equalizer.
4. Thus $\overline{R}(A^{\text{op}})$ is determined as a limit of $\{\overline{R}(A_i^\text{op})\}$.
5. I think 1~4 proves the uniqueness for objects up to isomorphism.
6. How do I construct the morphism $\overline{R}(f^\text{op}):B^\text{op}\rightarrow A^\text{op}$ for $f:A\rightarrow B$?
7. Is the proper left-exactness of this $\overline{R}$ easily proven?