A category $\smash {(\mathcal{C},\circ ^{\mathcal{C}},\mathbb {1}^{\mathcal{C}})}$ consists of:
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Objects. A class $\operatorname {\mathrm{Obj}}(\mathcal{C})$ of objects.
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Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, a class $\operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)$, called the class of morphisms of $\mathcal{C}$ from $A$ to $B$.
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Identities. For each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, a map of sets
\[ \mathbb {1}^{\mathcal{C}}_{A}\colon \mathrm{pt}\to \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,A), \]called the unit map of $\mathcal{C}$ at $A$, determining a morphism
\[ \operatorname {\mathrm{id}}_{A} \colon A \to A \]of $\mathcal{C}$, called the identity morphism of $A$.
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Composition. For each $A,B,C\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, a map of sets
\[ \circ ^{\mathcal{C}}_{A,B,C} \colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}(B,C)\times \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B) \to \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,C), \]called the composition map of $\mathcal{C}$ at $(A,B,C)$.
such that the following conditions are satisfied:
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Associativity. The diagram
commutes, i.e. for each composable triple $(f,g,h)$ of morphisms of $\mathcal{C}$, we have
\[ (f\circ g)\circ h = f\circ (g\circ h). \] -
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Left Unitality. The diagram
commutes, i.e. for each morphism $f\colon A\to B$ of $\mathcal{C}$, we have
\[ \operatorname {\mathrm{id}}_{B}\circ f=f. \] -
3.
Right Unitality. The diagram
commutes, i.e. for each morphism $f\colon A\to B$ of $\mathcal{C}$, we have
\[ f\circ \operatorname {\mathrm{id}}_{A}=f. \]
