diff --git a/tex/.vscode/ltex.dictionary.en-US.txt b/tex/.vscode/ltex.dictionary.en-US.txt
index f3b49f7..bf59a35 100644
--- a/tex/.vscode/ltex.dictionary.en-US.txt
+++ b/tex/.vscode/ltex.dictionary.en-US.txt
@@ -62,3 +62,9 @@ colimits
 colimit
 finitary
 Finitary
+cocone
+cocones
+Cocones
+coequalizer
+coequalizers
+pushouts
diff --git a/tex/main.pdf b/tex/main.pdf
index 0903fdb..aa80921 100644
Binary files a/tex/main.pdf and b/tex/main.pdf differ
diff --git a/tex/sections/02_categories.tex b/tex/sections/02_categories.tex
index 4952928..49f63e1 100644
--- a/tex/sections/02_categories.tex
+++ b/tex/sections/02_categories.tex
@@ -979,7 +979,7 @@ Dual to F-algebras the \emph{initial F-coalgebra} is trivial:
   % TODO example proof, needs some interesting functions introduced earlier.
 \end{example}
 
-\section{Limits}
+\section{Limits}\label{sec:limit}
 Limits are an abstraction of products and many other categorical concepts.
 \begin{definition}[Limit] We will need to introduce some related notions first.
   \begin{enumerate}
@@ -988,40 +988,40 @@ Limits are an abstraction of products and many other categorical concepts.
           \begin{itemize}
             \item an object $C \in \obj{\CC}$ called the \emph{apex} and
             \item a family of morphisms ${(f_d : C \to Dd)}_{d\in\obj{\CD}}$ such that
-                  % https://q.uiver.app/#q=WzAsMyxbMCwwLCJDIl0sWzAsMiwiRGQiXSxbMiwyLCJEZCciXSxbMCwxLCJmX2QiLDJdLFswLDIsImZfe2QnfSJdLFsxLDIsInUiXV0=
+                  % https://q.uiver.app/#q=WzAsMyxbMSwwLCJDIl0sWzAsMiwiRGQiXSxbMiwyLCJEZCciXSxbMCwxLCJmX2QiLDJdLFswLDIsImZfe2QnfSJdLFsxLDIsIkR1Il1d
                   \[\begin{tikzcd}
-                      C \\
+                      & C \\
                       \\
                       Dd && {Dd'}
-                      \arrow["{f_d}"', from=1-1, to=3-1]
-                      \arrow["{f_{d'}}", from=1-1, to=3-3]
-                      \arrow["u", from=3-1, to=3-3]
+                      \arrow["{f_d}"', from=1-2, to=3-1]
+                      \arrow["{f_{d'}}", from=1-2, to=3-3]
+                      \arrow["Du", from=3-1, to=3-3]
                     \end{tikzcd}\]
                   commutes for every $u : d \to d'$.
           \end{itemize}
-    \item A \emph{limit} of a diagram $D$ is a universal cone, i.e.\ a cone $(L, out_d)$ such that for every cone $(C, f_d)$ there exists a unique morphism $h : C \to L$ such that $out_d \comp h = f_d$ for all $d \in \obj{\CD}$:
-          % https://q.uiver.app/#q=WzAsMyxbMiwwLCJMIl0sWzIsMiwiRGQiXSxbMCwwLCJDIl0sWzAsMSwib3V0X2QiXSxbMiwwLCJoIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIsMSwiZl9kIiwyXV0=
+    \item A \emph{limit} of a diagram $D$ is a terminal cone, i.e.\ a cone $(L, out_d)$ such that for every cone $(C, f_d)$ there exists a unique morphism $h : C \to L$ such that $out_d \comp h = f_d$ for all $d \in \obj{\CD}$:
+          % https://q.uiver.app/#q=WzAsMyxbMiwwLCJMIl0sWzEsMiwiRGQiXSxbMCwwLCJDIl0sWzAsMSwib3V0X2QiXSxbMiwwLCJoIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIsMSwiZl9kIiwyXV0=
           \[\begin{tikzcd}
               C && L \\
               \\
-              && Dd
-              \arrow["{out_d}", from=1-3, to=3-3]
+              & Dd
+              \arrow["{out_d}", from=1-3, to=3-2]
               \arrow["h", dashed, from=1-1, to=1-3]
-              \arrow["{f_d}"', from=1-1, to=3-3]
+              \arrow["{f_d}"', from=1-1, to=3-2]
             \end{tikzcd}\]
   \end{enumerate}
 \end{definition}
 
 The notion of limit can be instantiated to many interesting notions:
 
-\begin{definition}[Terminal object (as limit)]
+\begin{definition}[Terminal Object (as Limit)]
   Let $\CD$ be the empty category $\emptyset$.
   Diagrams of $\CD$ are empty and thus cones consist of just the apex $C\in\obj{\CC}$.
   This also means that every object in $\CC$ forms a cone for this diagram.
   The limit $1 \in \obj{\CC}$ is thus the terminal object, since for any $C \in \obj{\CC}$ there is a unique morphism $\bang : C \to 1$.
 \end{definition}
 
-\begin{definition}[Product (as limit)]
+\begin{definition}[Product (as Limit)]
   Let $\CD$ be the discrete category with 2 elements. Diagrams $D$ are pairs $(A,B)$ of objects of $\CC$, cones are pairs of morphisms
   % https://q.uiver.app/#q=WzAsMyxbMSwwLCJDIl0sWzAsMSwiQSJdLFsyLDEsIkIiXSxbMCwxLCJmIiwyXSxbMCwyLCJnIl1d
   \[\begin{tikzcd}
@@ -1167,11 +1167,16 @@ The notion of limit can be instantiated to many interesting notions:
   A category $\CC$ is called \emph{complete} if every diagram in $\CC$ has a limit.
 \end{definition}
 
-\begin{proposition}
-  $\CC$ is complete iff $\CC$ has all products and equalizers, i.e.\ using products and equalizer one can construct arbitrary limits.
-\end{proposition}
+\begin{theorem}
+  The following are equivalent:
+  \begin{enumerate}
+    \item $\CC$ is complete
+    \item $\CC$ has (all) products and equalizers
+    \item $\CC$ has products and pullbacks
+  \end{enumerate}
+\end{theorem}
 \begin{proof}
-  % TODO proof that complete iff products and eq
+  % TODO
 \end{proof}
 
 \begin{definition}[Finitely Complete Category]
@@ -1193,4 +1198,157 @@ The notion of limit can be instantiated to many interesting notions:
 
 
 \section{Colimits}
-% TODO colimits
\ No newline at end of file
+Dual to the notion of limit is the notion of \emph{colimit}.
+
+\begin{definition}[Colimit]
+  Again, we will need a preliminary notion.
+  \begin{enumerate}
+    \item A \emph{cocone} of a diagram $D : \CD \to CC$ consists of
+          \begin{itemize}
+            \item an object $C \in \obj{\CC}$ that we call the \emph{foot} and
+            \item a family of morphisms ${(f_d : Dd \to C)}_{d \in \obj{\CD}}$ such that
+                  % https://q.uiver.app/#q=WzAsMyxbMSwyLCJDIl0sWzAsMCwiRGQiXSxbMiwwLCJEZCciXSxbMSwwLCJmX2QiLDJdLFsyLDAsImZfe2QnfSJdLFsxLDIsIkR1Il1d
+                  \[\begin{tikzcd}
+                      Dd && {Dd'} \\
+                      \\
+                      & C
+                      \arrow["{f_d}"', from=1-1, to=3-2]
+                      \arrow["{f_{d'}}", from=1-3, to=3-2]
+                      \arrow["Du", from=1-1, to=1-3]
+                    \end{tikzcd}\]
+                  commutes for every $u : d \to d'$.
+          \end{itemize}
+    \item A \emph{colimit} of a diagram $D$ is an initial cocone, i.e.\ a cocone $(L, in_d)$ such that for every cocone $(C, f_d)$ there exists a unique morphism $h : L \to C$ such that $h \comp in_d = f_d$ for all $d \in \obj{\CD}$:
+          % https://q.uiver.app/#q=WzAsMyxbMiwyLCJMIl0sWzEsMCwiRGQiXSxbMCwyLCJDIl0sWzEsMCwie2lufV9kIl0sWzAsMiwiXFxleGlzdHMhaCIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsxLDIsImZfZCIsMl1d
+          \[\begin{tikzcd}
+              & Dd \\
+              \\
+              C && L
+              \arrow["{{in}_d}", from=1-2, to=3-3]
+              \arrow["{\exists!h}"', dashed, from=3-3, to=3-1]
+              \arrow["{f_d}"', from=1-2, to=3-1]
+            \end{tikzcd}\]
+  \end{enumerate}
+\end{definition}
+Now we can instantiate the notion of colimit to the dual notions of \autoref{sec:limit}.
+
+\begin{definition}[Initial Object (as Colimit)]
+  Let $\CD$ be the empty category $\emptyset$.
+  Diagrams of $\CD$ are empty and thus cocones of these diagrams are equal to the cones, which consist of a single object.
+  The colimit $0 \in \obj{\CC}$ of such a diagram is thus the initial object, since for any $C\in\obj{\CC}$ there is a unique morphism $\cobang : 0 \to C$.
+\end{definition}
+
+\begin{definition}[Coproduct (as Colimit)]
+  Let $\CD$ be the discrete category with 2 elements. Cocones are pairs of morphisms:
+  % https://q.uiver.app/#q=WzAsMyxbMCwwLCJBIl0sWzIsMCwiQiJdLFsxLDEsIkMiXSxbMCwyLCJmIiwyXSxbMSwyLCJnIl1d
+  \[\begin{tikzcd}
+      A && B \\
+      & C
+      \arrow["f"', from=1-1, to=2-2]
+      \arrow["g", from=1-3, to=2-2]
+    \end{tikzcd}\]
+  and limits are exactly the coproducts:
+  % https://q.uiver.app/#q=WzAsNCxbMCwwLCJBIl0sWzIsMCwiQiJdLFsxLDEsIkErQiJdLFsxLDMsIkMiXSxbMCwyLCJcXGlubCJdLFsxLDIsIlxcaW5yIiwyXSxbMCwzLCJmIiwyLHsiY3VydmUiOjN9XSxbMSwzLCJnIiwwLHsiY3VydmUiOi0zfV0sWzIsMywiXFxleGlzdHMhW2YsZ10iLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=
+  \[\begin{tikzcd}
+      A && B \\
+      & {A+B} \\
+      \\
+      & C
+      \arrow["\inl", from=1-1, to=2-2]
+      \arrow["\inr"', from=1-3, to=2-2]
+      \arrow["f"', curve={height=18pt}, from=1-1, to=4-2]
+      \arrow["g", curve={height=-18pt}, from=1-3, to=4-2]
+      \arrow["{\exists![f,g]}"', dashed, from=2-2, to=4-2]
+    \end{tikzcd}\]
+\end{definition}
+
+\begin{definition}[Coequalizer]
+  Let $\CD$ be a category with two non-trivial and parallel morphisms $u,v : 1 \to 2$. Cocones are pairs of morphisms:
+  % https://q.uiver.app/#q=WzAsMyxbMCwwLCJBXzEiXSxbMiwwLCJBXzIiXSxbMSwyLCJDIl0sWzAsMSwiZiIsMCx7Im9mZnNldCI6LTF9XSxbMCwxLCJnIiwyLHsib2Zmc2V0IjoxfV0sWzAsMiwiYyIsMl0sWzEsMiwiZCJdXQ==
+  \[\begin{tikzcd}
+      {A_1} && {A_2} \\
+      \\
+      & C
+      \arrow["f", shift left, from=1-1, to=1-3]
+      \arrow["g"', shift right, from=1-1, to=1-3]
+      \arrow["c"', from=1-1, to=3-2]
+      \arrow["d", from=1-3, to=3-2]
+    \end{tikzcd}\]
+  The initial cocone is called a \emph{coequalizer} of $f$ and $g$:
+  % https://q.uiver.app/#q=WzAsNCxbMCwwLCJBXzEiXSxbMiwwLCJBXzIiXSxbMSwyLCJFIl0sWzEsNCwiQyJdLFswLDEsImYiLDAseyJvZmZzZXQiOi0xfV0sWzAsMSwiZyIsMix7Im9mZnNldCI6MX1dLFswLDJdLFsxLDIsImUiXSxbMCwzLCIiLDAseyJjdXJ2ZSI6M31dLFsxLDMsImMiLDAseyJjdXJ2ZSI6LTN9XSxbMiwzLCJcXGV4aXN0cyFoIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d
+  \[\begin{tikzcd}
+      {A_1} && {A_2} \\
+      \\
+      & E \\
+      \\
+      & C
+      \arrow["f", shift left, from=1-1, to=1-3]
+      \arrow["g"', shift right, from=1-1, to=1-3]
+      \arrow[from=1-1, to=3-2]
+      \arrow["e", from=1-3, to=3-2]
+      \arrow[curve={height=18pt}, from=1-1, to=5-2]
+      \arrow["c", curve={height=-18pt}, from=1-3, to=5-2]
+      \arrow["{\exists!h}", dashed, from=3-2, to=5-2]
+    \end{tikzcd}\]
+  where the unnamed morphisms are usually omitted.
+\end{definition}
+
+\begin{definition}[Regular Epimorphism]
+  An epimorphism is called \emph{regular} if it is also a coequalizer.
+\end{definition}
+
+\begin{lemma}
+  Every coequalizer is an epimorphism and thus a regular epimorphism.
+\end{lemma}
+\begin{proof}
+  By duality.
+\end{proof}
+\begin{proposition}
+  $e$ is a regular epimorphism and a monomorphism $\iff$ $e$ is an isomorphism.
+\end{proposition}
+\begin{proof}
+  By duality.
+\end{proof}
+
+\begin{definition}[Pushout]
+  % TODO
+\end{definition}
+
+\begin{lemma}
+  Colimits are unique up to isomorphism.
+\end{lemma}
+\begin{proof}
+  By duality.
+\end{proof}
+
+\begin{definition}
+  A category $\CC$ is called \emph{cocomplete}, if every diagram has a colimit in $\CC$.
+\end{definition}
+
+\begin{theorem} The following are equivalent:
+  \begin{enumerate}
+    \item $\CC$ is cocomplete
+    \item $\CC$ has (all) coproducts and coequalizers
+    \item $\CC$ has coproducts and pushouts
+  \end{enumerate}
+\end{theorem}
+\begin{proof}
+  % TODO
+\end{proof}
+
+\begin{definition}
+  A category $\CC$ is called \emph{finitely cocomplete}, if every finite diagram in $\CC$ has a limit.
+\end{definition}
+
+\begin{theorem}
+  The following are equivalent:
+  \begin{enumerate}
+    \item $\CC$ is finitely cocomplete
+    \item $\CC$ has finite coproducts and coequalizers
+    \item $\CC$ has finite coproducts and pushouts
+    \item $\CC$ has an initial object and pushouts
+  \end{enumerate}
+\end{theorem}
+\begin{proof}
+  % TODO
+\end{proof}
\ No newline at end of file