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