How does a map between inverse systems induce the inverse limit of its components?

Question

I encountered the following definition in the book Profinite Groups by Ribes and Zallesski:

Let $\{X_i, \varphi_{ij} \}$ and $\{ X_i', \varphi_{ij}' \}$ be inverse systems of topological spaces over the same directed poset $I$, and let $(X,\varphi_i)$ and $(X',\varphi_{ij}')$ be their corresponding inverse limits.
Assume $$\Theta: \{ X_i, \varphi_{ij}, I \} \to \{ X_i', \varphi_{ij}', I \}$$ is a map of inverse systems with components $\theta_i: X_i \to X_i'$.
Then the collection of compatible maps $\theta_i \varphi_i: X \to X_i'$ induces a continuous map $$ \varprojlim \Theta = \varprojlim_{i \in I}: X \to X'. $$

Question: In what regard is the map $\varprojlim \Theta$ induced? What does that actually mean?

I understand why $\theta_i \varphi_i$ are compatible and continuous. I assume that it has something to do with that.

Could you please explain that to me? Thank you!

Answer 1

This induced map exists by the very definition of an inverse limit as a topological space satisfying a universal property. Note that $X$ is a topological space, and each $\theta_i\varphi_i:X\to X_i'$ is continuous and compatible as you say. The universal property of inverse systems thus gives that there exists a unique continuous mapping, say $\tilde\Theta:X\to X'$, such that $\varphi'_i \tilde\Theta=\theta_i\varphi_i$ for all $i\in I$. So $\underleftarrow\lim \Theta$ is just this $\tilde \Theta$.

This is, however, a rather unsatisfactory explanation, as it seems to me that we are rather putting the cart before the horse in this very categorical argument. If possible it would be nice to construct $\underleftarrow\lim \Theta$ and then show that it satisfies the universal property. This would be ideal if we want to examine explicit examples. In the construction of the inverse limit we see that $X=\{\{x_i\}_{i\in I}:\varphi_{ij}x_i=x_j~\text{for all}~i\succeq j\}$ is a subspace of $\prod_{i\in I} X_i$ (similarly for $X'$). Therefore it is natural to define $\underleftarrow\lim \Theta:X\to \prod_{i\in I}X_i'$ by $$\underleftarrow\lim \Theta(\{x_i\}_{i\in I})=\{\theta_i(x_i)\}_{i\in I}.$$ It is a nice exercise to show that the image of $\underleftarrow\lim \Theta$ so defined is indeed contained in $X'$, and $\underleftarrow\lim \Theta$ is continuous, and it satisfies the universal property. These properties all follow from the continuity and compatibility of the collection $\{\theta_i\}_{i\in I}$.