Final corrections (#162)

* Wow. Such corrections. 🐶

* Other corrections for Chapter 30.

* Update errata.

errata.md

@@ -8,6 +8,14 @@
 
 * [#157](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/157) - Adding paragraph indent
 
+### 12. Limits and Colimits
+
+* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical error
+
+### 14. Representable Functors
+
+* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical error
+
 ### 18. Adjunctions
 
 * [#160](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/160) - Fix spelling of "counit"
@@ -20,6 +28,11 @@
 ### 20. Monads - Programmer's Definition
 
 * [#160](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/160) - Fix grammatical error
+* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical error
+
+### 22. Monads Categorically
+
+* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical error
 
 ### 23. Comonads
 
@@ -30,6 +43,7 @@
 * [#158](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/158) - fixed incorrect typesetting of `set`
 * [#159](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/159) - fixed incorrect typesetting of category terms
 * [#160](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/160) - Fix spelling of "counit" and "morphisms", fix subscript spacing
+* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical errors
 
 ### 26. Ends and Coends
 
@@ -43,7 +57,13 @@
 ### 28. Enriched Categories
 
 * [#160](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/160) - Fix subscript spacing
+* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical error
+
+### 29. Topoi
+
+* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical error
 
 ### 30. Lawvere Theories
 
 * [#160](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/160) - Fix spelling of "coequalizer"
+* [#162](https://github.com/hmemcpy/milewski-ctfp-pdf/pull/162) - Fix grammatical errors and a typesetting error

src/content/2.2/Limits and Colimits.tex

@@ -83,7 +83,7 @@ identities) in $\cat{2}$.
 A generalization of this construction to categories other than
 $\cat{2}$ --- ones that, for instance, contain non-trivial morphisms
 --- will impose naturality conditions on the transformation between
-$\Delta_c$ and $D$. We call such transformation a \emph{cone},
+$\Delta_c$ and $D$. We call such a transformation a \emph{cone},
 because the image of $\Delta$ is the apex of a cone/pyramid whose sides are
 formed by the components of the natural transformation. The image of $D$
 forms the base of the cone.

src/content/2.4/Representable Functors.tex

@@ -139,7 +139,7 @@ $\cat{C}$ as sets and functions in $\Set$.
 The functor $\cat{C}(a, -)$ itself is sometimes called representable.
 More generally, any functor $F$ that is naturally isomorphic to
 the hom-functor, for some choice of $a$, is called
-\newterm{representable}. Such functor must necessarily be
+\newterm{representable}. Such a functor must necessarily be
 $\Set$-valued, since $\cat{C}(a, -)$ is.
 
 I said before that we often think of isomorphic sets as identical. More

src/content/3.12/Enriched Categories.tex

@@ -299,7 +299,7 @@ preorder). The monoidal structure is given by addition, with zero
 serving as the unit object. In other words, the tensor product of two
 numbers is their sum.
 
-A metric space is a category enriched over such monoidal category. A
+A metric space is a category enriched over such a monoidal category. A
 hom-object $\cat{C}(a, b)$ from object $a$ to $b$ is a
 non-negative (possibly infinite) number that we will call the distance
 from $a$ to $b$. Let's see what we get for identity and

src/content/3.13/Topoi.tex

@@ -9,7 +9,7 @@ of Haskell's type system with dependent types, or the exploration on
 homotopy type theory in programming.
 
 So far I've been casually identifying types with \emph{sets} of values.
-This is not strictly correct, because such approach doesn't take into
+This is not strictly correct, because such an approach doesn't take into
 account the fact that, in programming, we \emph{compute} values, and the
 computation is a process that takes time and, in extreme cases, might
 not terminate. Divergent computations are part of every Turing-complete

src/content/3.14/Lawvere Theories.tex

@@ -139,7 +139,7 @@ $1 \times 1$ (or $1^2$) in $\cat{L}$. In this sense, the category
 $\cat{F}$ behaves like the logarithm of $\cat{L}$.
 
 Among morphisms in $\cat{L}$ we have those transferred by the functor
-$I_{\cat{L}}$ from $\cat{F}$. They play structural role in $\cat{L}$. In
+$I_{\cat{L}}$ from $\cat{F}$. They play a structural role in $\cat{L}$. In
 particular coproduct injections $i_k$ become product projections
 $p_k$. A useful intuition is to imagine the projection:
 \[p_k \Colon 1^n \to 1\]
@@ -219,7 +219,7 @@ The preservation of products by models means that the image of
 $M$ in $\Set$ is a sequence of sets generated by powers of
 the set $M\ 1$ --- the image of the object $1$ from
 $\cat{L}$. Let's call this set $a$. (This set is sometimes
-called a \emph{sort}, and such algebra is called \newterm{single-sorted}. There
+called a \emph{sort}, and such an algebra is called \newterm{single-sorted}. There
 exist generalizations of Lawvere theories to multi-sorted algebras.) In
 particular, binary operations from $\cat{L}$ are mapped to functions:
 \[a \times a \to a\]
@@ -252,7 +252,7 @@ The functors that define models form a category of models,
 $\cat{Mod}(\cat{L}, \Set)$, with natural transformations as morphisms.
 
 Consider a model for the trivial Lawvere category
-$\Fop$. Such model is completely determined by
+$\Fop$. Such a model is completely determined by
 its value at $1$, $M\ 1$. Since $M\ 1$ can be any
 set, there are as many of these models as there are sets in
 $\Set$. Moreover, every morphism in $\cat{Mod}(\Fop, \Set)$ (a
@@ -279,7 +279,7 @@ elegant construction.
 Every monoid must have a unit, so we have to have a special morphism
 $\eta$ in $\cat{L}_{\cat{Mon}}$ that goes from $0$ to
 $1$. Notice that there can be no corresponding morphism in
-$\cat{F}$. Such morphism would go in the opposite direction, from
+$\cat{F}$. Such a morphism would go in the opposite direction, from
 $1$ to $0$ which, in $\cat{FinSet}$, would be a function
 from the singleton set to the empty set. No such function exists.
 
@@ -334,7 +334,7 @@ $\cat{L}$.
 Another way of deriving $U$ is by exploiting the fact that
 $\Fop$ is the initial object in $\cat{Law}$. It
 means that, for any Lawvere theory $\cat{L}$, there is a unique
-functor $\Fop \to L$. This functor induces the
+functor $\Fop \to \cat{L}$. This functor induces the
 opposite functor on models (since models are functors \emph{from}
 theories to sets):
 \[\cat{Mod}(\cat{L}, \Set) \to \cat{Mod}(\Fop, \Set)\]

src/content/3.4/Monads - Programmer's Definition.tex

@@ -85,7 +85,7 @@ We have previously arrived at the
 \hyperref[kleisli-categories]{writer
 monad} by embellishing regular functions. The particular embellishment
 was done by pairing their return values with strings or, more generally,
-with elements of a monoid. We can now recognize that such embellishment
+with elements of a monoid. We can now recognize that such an embellishment
 is a functor:
 
 \src{code/haskell/snippet02.hs}

src/content/3.6/Monads Categorically.tex

@@ -524,7 +524,7 @@ This is indeed the implementation of \code{join} for the
 
 It turns out that not only every adjunction gives rise to a monad, but
 the converse is also true: every monad can be factorized into a
-composition of two adjoint functors. Such factorization is not unique
+composition of two adjoint functors. Such a factorization is not unique
 though.
 
 We'll talk about the other endofunctor $L \circ R$ in the next

src/content/3.9/Algebras for Monads.tex

@@ -15,7 +15,7 @@ every adjunction \hyperref[monads-categorically]{defines
 a monad} (and a comonad). The question is: Can every monad (comonad) be
 derived from an adjunction? The answer is positive. There is a whole
 family of adjunctions that generate a given monad. I'll show you two
-such adjunction.
+such adjunctions.
 Let's review the definitions. A monad is an endofunctor $m$
 equipped with two natural transformations that satisfy some coherence
 conditions. The components of these transformations at $a$ are:
@@ -165,7 +165,7 @@ the pair $(T\ a, \mu_a)$. So in order to define the component of
 the counit $\varepsilon$ at $(a, f)$, we need the right morphism in
 the Eilenberg-Moore category, or a homomorphism of T-algebras:
 \[(T\ a, \mu_a) \to (a, f)\]
-Such homomorphism should map the carrier $T\ a$ to $a$.
+Such a homomorphism should map the carrier $T\ a$ to $a$.
 Let's just resurrect the forgotten evaluator $f$. This time we'll
 use it as a homomorphism of T-algebras. Indeed, the same commuting
 diagram that makes $f$ a T-algebra may be re-interpreted to show