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### 5 Category 2-Cells

#### 5.1 Attributes for the Type of 2-Cells

##### 5.1-1 Source
 ‣ Source( c ) ( attribute )

Returns: a morphism

The argument is a 2-cell c: \alpha \rightarrow \beta. The output is its source \alpha.

##### 5.1-2 Range
 ‣ Range( c ) ( attribute )

Returns: a morphism

The argument is a 2-cell c: \alpha \rightarrow \beta. The output is its range \beta.

#### 5.2 Identity 2-Cell and Composition of 2-Cells

##### 5.2-1 IdentityTwoCell
 ‣ IdentityTwoCell( alpha ) ( attribute )

Returns: a 2-cell

The argument is a morphism \alpha. The output is its identity 2-cell \mathrm{id}_{\alpha}: \alpha \rightarrow \alpha.

 ‣ AddIdentityTwoCell( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IdentityTwoCell. F: \alpha \mapsto \mathrm{id}_{\alpha}.

##### 5.2-3 HorizontalPreCompose
 ‣ HorizontalPreCompose( c, d ) ( operation )

Returns: a 2-cell

The arguments are two 2-cells c: \alpha \rightarrow \beta, d: \gamma \rightarrow \delta between morphisms \alpha, \beta: a \rightarrow b and \gamma, \delta: b \rightarrow c. The output is their horizontal composition d \ast c: (\gamma \circ \alpha) \rightarrow (\delta \circ \beta).

 ‣ AddHorizontalPreCompose( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation HorizontalPreCompose. F: (c,d) \mapsto d \ast c.

##### 5.2-5 HorizontalPostCompose
 ‣ HorizontalPostCompose( d, c ) ( operation )

Returns: a 2-cell

The arguments are two 2-cells d: \gamma \rightarrow \delta, c: \alpha \rightarrow \beta between morphisms \alpha, \beta: a \rightarrow b and \gamma, \delta: b \rightarrow c. The output is their horizontal composition d \ast c: (\gamma \circ \alpha) \rightarrow (\delta \circ \beta).

 ‣ AddHorizontalPostCompose( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation HorizontalPostCompose. F: (d,c) \mapsto d \ast c.

##### 5.2-7 VerticalPreCompose
 ‣ VerticalPreCompose( c, d ) ( operation )

Returns: a 2-cell

The arguments are two 2-cells c: \alpha \rightarrow \beta, d: \beta \rightarrow \gamma between morphisms \alpha, \beta, \gamma: a \rightarrow b. The output is their vertical composition d \circ c: \alpha \rightarrow \gamma.

 ‣ AddVerticalPreCompose( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation VerticalPreCompose. F: (c,d) \mapsto d \circ c.

##### 5.2-9 VerticalPostCompose
 ‣ VerticalPostCompose( d, c ) ( operation )

Returns: a 2-cell

The arguments are two 2-cells d: \beta \rightarrow \gamma, c: \alpha \rightarrow \beta between morphisms \alpha, \beta, \gamma: a \rightarrow b. The output is their vertical composition d \circ c: \alpha \rightarrow \gamma.

 ‣ AddVerticalPostCompose( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation VerticalPostCompose. F: (d,c) \mapsto d \circ c.

#### 5.3 Well-Definedness for 2-Cells

##### 5.3-1 IsWellDefinedForTwoCells
 ‣ IsWellDefinedForTwoCells( c ) ( operation )

Returns: a boolean

The argument is a 2-cell c. The output is true if c is well-defined, otherwise the output is false.

 ‣ AddIsWellDefinedForTwoCells( C, F ) ( operation )
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsWellDefinedForTwoCells. F: c \mapsto \mathtt{IsWellDefinedForMorphisms}( c ).