Direct limit, inverse limit and Spec

Question

The set-up

1. $k$ is a field
2. $T_i = \operatorname{Spec} A_i$ is an inverse system of affine $k$-schemes, where $i<j$ if $\operatorname{Spec} A_j \subset \operatorname{Spec} A_i$ (inclusion).
3. $X$ is a $k$-variety (hence locally of finite presentation).

Then, by this porposition, we have

$$ \operatorname{Mor}(\varprojlim_i T_i, X) = \varinjlim_i \operatorname{Mor}(T_i, X)$$

The confusion

In the proof, we use the direct limit $A = \varinjlim_i A_i$ and then use it to create the inverse system $T_i$ using the fact that $\varphi : A \to B$ induces $\operatorname{Spec} \phi: \operatorname{Spec} B \to \operatorname{Spec} A$ via contraction. This should work sice the functor between rings and schemes is contravariant. However, the converse relation isn't true: spec of inverse limit of rings is not same as direct limit of spec of rings.

The question

How can we get the direct system $(A_i, \varphi_{ij})$ from the above inverse system $(T_i, f_{ij})$? In particular, what is the meaning of this statement

$$\varprojlim_i \operatorname{Spec} A_i = \operatorname{Spec} \varinjlim_i A_i$$

Edit

I have realized that this was a dumb question. The scheme morphism $\operatorname{Spec} A \to \operatorname{Spec} B$ by definition gives the ring homomorphism $B \to A$. Moreover, as commented below by Martin Brandenburg and Zhen Lin on the linked question, spectrum commutes with filtered colimits (direct limit).