Definition 4.29.1. A (strict) $2$-category $\mathcal{C}$ consists of the following data

1. A set of objects $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

2. For each pair $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ a category $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$. The objects of $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$ will be called $1$-morphisms and denoted $F : x \to y$. The morphisms between these $1$-morphisms will be called $2$-morphisms and denoted $t : F' \to F$. The composition of $2$-morphisms in $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$ will be called vertical composition and will be denoted $t \circ t'$ for $t : F' \to F$ and $t' : F'' \to F'$.

3. For each triple $x, y, z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ a functor

$(\circ , \star ) : \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(y, z) \times \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, z).$

The image of the pair of $1$-morphisms $(F, G)$ on the left hand side will be called the composition of $F$ and $G$, and denoted $F\circ G$. The image of the pair of $2$-morphisms $(t, s)$ will be called the horizontal composition and denoted $t \star s$.

These data are to satisfy the following rules:

1. The set of objects together with the set of $1$-morphisms endowed with composition of $1$-morphisms forms a category.

2. Horizontal composition of $2$-morphisms is associative.

3. The identity $2$-morphism $\text{id}_{\text{id}_ x}$ of the identity $1$-morphism $\text{id}_ x$ is a unit for horizontal composition.

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