As explained in the text following Definition \ref{definition-system-over-poset}, we may view preordered sets as categories and systems as functors. Throughout the proof, we will freely shift between these two points of view. We prove the first statement by constructing a category $\mathcal{I}_0$, corresponding to a directed set\footnote{In fact, our construction will produce a directed partially ordered set.}, and a cofinal functor $M_0 : \mathcal{I}_0 \to \mathcal{I}$. Then, by Lemma \ref{lemma-cofinal}, the colimit of a diagram $M : \mathcal{I} \to \mathcal{C}$ coincides with the colimit of the diagram $M \circ M_0 | \mathcal{I}_0 \to \mathcal{C}$, from which the statement follows. The second statement is dual to the first and may be proved by interpreting a limit in $\mathcal{C}$ as a colimit in $\mathcal{C}^{opp}$. We omit the details. \medskip\noindent A category $\mathcal{F}$ is called {\em finitely generated} if there exists a finite set $F$ of arrows in $\mathcal{F}$, such that each arrow in $\mathcal{F}$ may be obtained by composing arrows from $F$. In particular, this implies that $\mathcal{F}$ has finitely many objects. We start the proof by reducing to the case when $\mathcal{I}$ has the property that every finitely generated subcategory of $\mathcal{I}$ may be extended to a finitely generated subcategory with a unique final object. \medskip\noindent Let $\omega$ denote the directed set of finite ordinals, which we view as a filtered category. It is easy to verify that the product category $\mathcal{I}\times \omega$ is also filtered, and the projection $\Pi : \mathcal{I} \times \omega \to \mathcal{I}$ is cofinal. \medskip\noindent Now let $\mathcal{F}$ be any finitely generated subcategory of $\mathcal{I}\times \omega$. By using the axioms of a filtered category and a simple induction argument on a finite set of generators of $\mathcal{F}$, we may construct a cocone $(\{f_i\}, i_\infty)$ in $\mathcal{I}$ for the diagram $\mathcal{F} \to \mathcal{I}$. That is, a morphism $f_i : i \to i_\infty$ for every object $i$ in $\mathcal{F}$ such that for each arrow $f : i \to i'$ in $\mathcal{F}$ we have $f_i = f\circ f_{i'}$. We can also choose $i_\infty$ such that there are no arrows from $i_\infty$ to an object in $\mathcal{F}$. This is possible since we may always post-compose the arrows $f_i$ with an arrow which is the identity on the $\mathcal{I}$-component and strictly increasing on the $\omega$-component. Now let $\mathcal{F}^+$ denote the category consisting of all objects and arrows in $\mathcal{F}$ together with the object $i_\infty$, the identity arrow $\text{id}_{i_\infty}$ and the arrows $f_i$. Since there are no arrows from $i_\infty$ in $\mathcal{F}^+$ to any object of $\mathcal{F}$, the arrow set in $\mathcal{F}^+$ is closed under composition, so $\mathcal{F}^+$ is indeed a category. By construction, it is a finitely generated subcategory of $\mathcal{I}$ which has $i_\infty$ as unique final object. Since, by Lemma \ref{lemma-cofinal}, the colimit of  any diagram $M : \mathcal{I} \to \mathcal{C}$ coincides with the colimit of $M\circ\Pi$ , this gives the desired reduction. \medskip\noindent The set of all finitely generated subcategories of $\mathcal{I}$ with a unique final object is naturally ordered by inclusion. We take $\mathcal{I}_0$ to be the category corresponding to this set. We also have a functor $M_0 : \mathcal{I}_0 \to \mathcal{I}$, which takes an arrow $\mathcal{F} \subset \mathcal{F'}$ in $\mathcal{I}_0$ to the unique map from the final object of $\mathcal{F}$ to the final object of $\mathcal{F}'$. Given any two finitely generated subcategories of $\mathcal{I}$, the category generated by these two categories is also finitely generated. By our assumption on $\mathcal{I}$, it is also contained in a finitely generated subcategory of $\mathcal{I}$ with a unique final object. This shows that $\mathcal{I}_0$ is directed. \medskip\noindent Finally, we verify that $M_0$ is cofinal. Since any object of $\mathcal{I}$ is the final object in the subcategory consisting of only that object and its identity arrow, the functor $M_0$ is surjective on objects. In particular, Condition (1) of Definition \ref{definition-cofinal} is satisfied. Given an object $i$ of $\mathcal{I}$, $\mathcal{F}_1, \mathcal{F}_2$ in $\mathcal{I}_0$ and maps $\varphi_1 : i \to M_0(\mathcal{F}_1)$ and $\varphi_2 : i \to M_0(\mathcal{F}_2)$ in $\mathcal{I}$, we can take $\mathcal{F}_{12}$ to be a finitely generated category with a unique final object containing $\mathcal{F}_1$, $\mathcal{F}_2$ and the morphisms $\varphi_1, \varphi_2$. The resulting diagram commutes $$ \xymatrix{ & M_0(\mathcal{F}_{12}) & \\ M_0(\mathcal{F}_{1}) \ar[ru] & & M_0(\mathcal{F}_{2}) \ar[lu] \\ & i \ar[lu] \ar[ru] } $$ since it lives in the category $\mathcal{F}_{12}$ and $M_0(\mathcal{F}_{12})$ is final in this category. Hence also Condition (2) is satisfied, which concludes the proof.