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[PC] X-COM UFO Defense Gold Edition [by Trapacer] CPY

[PC] X-COM UFO Defense Gold Edition [by Trapacer] CPY
X-COM: UFO Defense Gold Edition X-COM has been downloaded from over 17 million. [PC] X-COM UFO Defense Gold Edition [by trapacer] CPYQ:

Proving that the colimit of a free inverse limit is free

On page 5 of these notes about categories and fibered categories, the following is mentioned:

Lemma. Let $\{A_i\}_{i \in I}$ be a family of objects in a category $\mathcal{C}$ and let $X$ be an object in $\mathcal{C}$. Let $\varprojlim_{i \in I} A_i$ be the colimit of this system of maps. Then, $A := \varprojlim_{i \in I} A_i$ is the free object on the family $\{A_i\}_{i \in I}$.

I have seen proofs of this lemma before. However, I’ve been unable to make sense of this proof given in the above notes. The idea is to show that $A$ is the free object on $\{A_i\}_{i \in I}$ by showing that the universal property characterizing free objects (namely a canonical morphism of any $B$ to $A$ such that the universal property holds for every $f: B \to A$) is given by:
$$
\varphi: \varinjlim_{i \in I} A_i \to A,
$$
where $\varphi(a) = a_0$ for some $a_0 \in A$, since $A$ is the colimit of $\{A_i\}_{i \in I}$. I know how to prove that the universal property holds for every $\varphi: B \to A$ satisfying the conditions given in the lemma. However, I don’t understand how exactly one can use the universality of $\varphi$ to show that the morphism $\varphi$ must indeed be the universal map $A \to B$.
Any help is appreciated. Thanks.

A:

The idea is that if you have a sequence of maps $X \stackrel{f_n}{\longrightarrow}A \stackrel{f_2}{

https://colab.research.google.com/drive/18DhbekRHL5S__gbx9cvoZ4D5EpQYTC5S
https://colab.research.google.com/drive/1043h76mMEapvDTpEGjLAtsIEd2TnNHgV
https://colab.research.google.com/drive/18PTocRs6W6UfjfMi0Umu2Bf9EXLmuWrV
https://colab.research.google.com/drive/1igMFmrF8ZH_MjLl252GCQRIwEW75zDKe
https://colab.research.google.com/drive/1f0lVTjfeWM-VqLXqzrJM2zB6_ClfctQe

[PC] X-COM UFO Defense Gold Edition [by Trapacer] CPY
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